Write The Quadratic Function In Vertex Form Whose Graph Has The Given Vertex And Passes Through The Given Point. Vertex: { (2,3)$}$ Point: { (4,15)$}$ { F(x) = $}$

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Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. The vertex form of a quadratic function is a powerful tool for analyzing and graphing these functions. In this article, we will discuss how to write a quadratic function in vertex form given a vertex and a point on the graph.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k

where (h,k)(h, k) is the vertex of the parabola, and aa is a constant that determines the direction and width of the parabola.

Given Vertex and Point

We are given the vertex (2,3)(2, 3) and the point (4,15)(4, 15). Our goal is to write the quadratic function in vertex form that passes through these two points.

Step 1: Write the Vertex Form with the Given Vertex

Since we are given the vertex (2,3)(2, 3), we can write the vertex form of the quadratic function as:

f(x)=a(x−2)2+3f(x) = a(x - 2)^2 + 3

Step 2: Use the Given Point to Find the Value of a

We are given the point (4,15)(4, 15), which lies on the graph of the quadratic function. We can substitute this point into the vertex form of the function to find the value of aa.

15=a(4−2)2+315 = a(4 - 2)^2 + 3

Simplifying the equation, we get:

15=a(2)2+315 = a(2)^2 + 3

15=4a+315 = 4a + 3

Subtracting 3 from both sides, we get:

12=4a12 = 4a

Dividing both sides by 4, we get:

a=3a = 3

Step 3: Write the Quadratic Function in Vertex Form

Now that we have found the value of aa, we can write the quadratic function in vertex form:

f(x)=3(x−2)2+3f(x) = 3(x - 2)^2 + 3

Graphing the Quadratic Function

To graph the quadratic function, we can use the vertex form of the function. Since the vertex is (2,3)(2, 3), we know that the graph of the function will be a parabola that opens upward and has its vertex at (2,3)(2, 3). The value of aa determines the width of the parabola, and in this case, a=3a = 3, so the parabola will be relatively wide.

Conclusion

In this article, we discussed how to write a quadratic function in vertex form given a vertex and a point on the graph. We used the given vertex and point to find the value of aa and then wrote the quadratic function in vertex form. We also graphed the quadratic function using the vertex form of the function. This is a powerful tool for analyzing and graphing quadratic functions, and it has numerous applications in various fields.

Example Problems

Problem 1

Write the quadratic function in vertex form whose graph has the given vertex and passes through the given point.

Vertex: (1,2)(1, 2) Point: (3,10)(3, 10)

Solution

We can write the vertex form of the quadratic function as:

f(x)=a(x−1)2+2f(x) = a(x - 1)^2 + 2

Substituting the point (3,10)(3, 10) into the vertex form of the function, we get:

10=a(3−1)2+210 = a(3 - 1)^2 + 2

Simplifying the equation, we get:

10=a(2)2+210 = a(2)^2 + 2

10=4a+210 = 4a + 2

Subtracting 2 from both sides, we get:

8=4a8 = 4a

Dividing both sides by 4, we get:

a=2a = 2

Therefore, the quadratic function in vertex form is:

f(x)=2(x−1)2+2f(x) = 2(x - 1)^2 + 2

Problem 2

Write the quadratic function in vertex form whose graph has the given vertex and passes through the given point.

Vertex: (−2,1)(-2, 1) Point: (0,5)(0, 5)

Solution

We can write the vertex form of the quadratic function as:

f(x)=a(x+2)2+1f(x) = a(x + 2)^2 + 1

Substituting the point (0,5)(0, 5) into the vertex form of the function, we get:

5=a(0+2)2+15 = a(0 + 2)^2 + 1

Simplifying the equation, we get:

5=a(2)2+15 = a(2)^2 + 1

5=4a+15 = 4a + 1

Subtracting 1 from both sides, we get:

4=4a4 = 4a

Dividing both sides by 4, we get:

a=1a = 1

Therefore, the quadratic function in vertex form is:

f(x)=1(x+2)2+1f(x) = 1(x + 2)^2 + 1

f(x) = (x + 2)^2 + 1$<br/> # **Quadratic Functions in Vertex Form: Q&A** =============================================

Introduction

In our previous article, we discussed how to write a quadratic function in vertex form given a vertex and a point on the graph. In this article, we will answer some frequently asked questions about quadratic functions in vertex form.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by:

f(x)=a(x−h)2+k</span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>isthevertexoftheparabola,and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>isaconstantthatdeterminesthedirectionandwidthoftheparabola.</p><h2><strong>Q:HowdoIfindthevalueofainthevertexformofaquadraticfunction?</strong></h2><hr><p>A:Tofindthevalueof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>,youcanusethegivenpointonthegraphandsubstituteitintothevertexformofthefunction.Then,solvefor<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>.</p><h2><strong>Q:WhatifIhaveaquadraticfunctioninstandardform,howcanIconvertittovertexform?</strong></h2><hr><p>A:Toconvertaquadraticfunctionfromstandardformtovertexform,youcancompletethesquare.Thisinvolvesrewritingthequadraticfunctionintheform:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mi>a</mi><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow><annotationencoding="application/x−tex">f(x)=a(x−h)2+k</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">h</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span></span></span></span></span></p><h2><strong>Q:HowdoIgraphaquadraticfunctioninvertexform?</strong></h2><hr><p>A:Tographaquadraticfunctioninvertexform,youcanusethevertexformofthefunction.Sincethevertexis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>,youknowthatthegraphofthefunctionwillbeaparabolathatopensupwardordownwardandhasitsvertexat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>.Thevalueof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>determinesthewidthoftheparabola.</p><h2><strong>Q:CanIusethevertexformofaquadraticfunctiontofindthex−interceptsofthegraph?</strong></h2><hr><p>A:Yes,youcanusethevertexformofaquadraticfunctiontofindthex−interceptsofthegraph.Todothis,setthefunctionequaltozeroandsolvefor<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotationencoding="application/x−tex">x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span></span></span></span>.</p><h2><strong>Q:HowdoIusethevertexformofaquadraticfunctiontofindthemaximumorminimumvalueofthefunction?</strong></h2><hr><p>A:Tofindthemaximumorminimumvalueofaquadraticfunction,youcanusethevertexformofthefunction.Sincethevertexis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>,youknowthatthemaximumorminimumvalueofthefunctionis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotationencoding="application/x−tex">k</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span></span></span></span>.</p><h2><strong>Q:CanIusethevertexformofaquadraticfunctiontofindtheequationoftheaxisofsymmetry?</strong></h2><hr><p>A:Yes,youcanusethevertexformofaquadraticfunctiontofindtheequationoftheaxisofsymmetry.Theaxisofsymmetryisaverticallinethatpassesthroughthevertexoftheparabola,anditsequationisgivenby<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>h</mi></mrow><annotationencoding="application/x−tex">x=h</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">h</span></span></span></span>.</p><h2><strong>Q:HowdoIusethevertexformofaquadraticfunctiontofindtheequationoftheparabola?</strong></h2><hr><p>A:Tofindtheequationoftheparabola,youcanusethevertexformofthefunction.Sincethevertexis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>,youknowthattheequationoftheparabolaisgivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mi>a</mi><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow><annotationencoding="application/x−tex">f(x)=a(x−h)2+k</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">h</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span></span></span></span></span></p><h2><strong>ExampleProblems</strong></h2><hr><h3><strong>Problem1</strong></h3><p>Findthevalueof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotationencoding="application/x−tex">a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span></span></span></span>inthevertexformofthequadraticfunction:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mi>a</mi><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">f(x)=a(x−2)2+3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span></span></p><p>Givenpoint:<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mn>4</mn><moseparator="true">,</mo><mn>15</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(4,15)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">4</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">15</span><spanclass="mclose">)</span></span></span></span></p><h3><strong>Solution</strong></h3><p>Substitutingthepoint<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mn>4</mn><moseparator="true">,</mo><mn>15</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(4,15)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">4</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">15</span><spanclass="mclose">)</span></span></span></span>intothevertexformofthefunction,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>15</mn><mo>=</mo><mi>a</mi><mostretchy="false">(</mo><mn>4</mn><mo>−</mo><mn>2</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">15=a(4−2)2+3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">15</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mopen">(</span><spanclass="mord">4</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span></span></p><p>Simplifyingtheequation,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>15</mn><mo>=</mo><mi>a</mi><mostretchy="false">(</mo><mn>2</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">15=a(2)2+3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">15</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal">a</span><spanclass="mopen">(</span><spanclass="mord">2</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span></span></p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>15</mn><mo>=</mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">15=4a+3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">15</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">4</span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span></span></p><p>Subtracting3frombothsides,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>12</mn><mo>=</mo><mn>4</mn><mi>a</mi></mrow><annotationencoding="application/x−tex">12=4a</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">12</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span><spanclass="mordmathnormal">a</span></span></span></span></span></p><p>Dividingbothsidesby4,weget:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>a</mi><mo>=</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">a=3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">a</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span></span></p><h3><strong>Problem2</strong></h3><p>Findtheequationoftheaxisofsymmetryoftheparabolagivenby:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>2</mn><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><annotationencoding="application/x−tex">f(x)=2(x−3)2+2</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">3</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span></span></span></span></span></p><h3><strong>Solution</strong></h3><p>Theequationoftheaxisofsymmetryisgivenby<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>h</mi></mrow><annotationencoding="application/x−tex">x=h</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">h</span></span></span></span>,where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>isthevertexoftheparabola.Inthiscase,thevertexis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mn>3</mn><moseparator="true">,</mo><mn>2</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(3,2)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">3</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span></span></span></span>,sotheequationoftheaxisofsymmetryis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><annotationencoding="application/x−tex">x=3</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">3</span></span></span></span></span></p><h3><strong>Problem3</strong></h3><p>Findthemaximumvalueofthequadraticfunction:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>f</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo><mo>=</mo><mn>2</mn><mostretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><msup><mostretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><annotationencoding="application/x−tex">f(x)=2(x−1)2+4</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1.1141em;vertical−align:−0.25em;"></span><spanclass="mord">1</span><spanclass="mclose"><spanclass="mclose">)</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8641em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight">2</span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span></span></span></span></span></p><h3><strong>Solution</strong></h3><p>Themaximumvalueofthefunctionisgivenby<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotationencoding="application/x−tex">k</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span></span></span></span>,where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mi>h</mi><moseparator="true">,</mo><mi>k</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(h,k)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mordmathnormal">h</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mclose">)</span></span></span></span>isthevertexoftheparabola.Inthiscase,thevertexis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mostretchy="false">(</mo><mn>1</mn><moseparator="true">,</mo><mn>4</mn><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">(1,4)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mopen">(</span><spanclass="mord">1</span><spanclass="mpunct">,</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mord">4</span><spanclass="mclose">)</span></span></span></span>,sothemaximumvalueofthefunctionis:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>k</mi><mo>=</mo><mn>4</mn></mrow><annotationencoding="application/x−tex">k=4</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03148em;">k</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">4</span></span></span></span></span></p>f(x) = a(x - h)^2 + k </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> is the vertex of the parabola, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> is a constant that determines the direction and width of the parabola.</p> <h2><strong>Q: How do I find the value of a in the vertex form of a quadratic function?</strong></h2> <hr> <p>A: To find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>, you can use the given point on the graph and substitute it into the vertex form of the function. Then, solve for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>.</p> <h2><strong>Q: What if I have a quadratic function in standard form, how can I convert it to vertex form?</strong></h2> <hr> <p>A: To convert a quadratic function from standard form to vertex form, you can complete the square. This involves rewriting the quadratic function in the form:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f(x) = a(x - h)^2 + k </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span></p> <h2><strong>Q: How do I graph a quadratic function in vertex form?</strong></h2> <hr> <p>A: To graph a quadratic function in vertex form, you can use the vertex form of the function. Since the vertex is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>, you know that the graph of the function will be a parabola that opens upward or downward and has its vertex at <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>. The value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> determines the width of the parabola.</p> <h2><strong>Q: Can I use the vertex form of a quadratic function to find the x-intercepts of the graph?</strong></h2> <hr> <p>A: Yes, you can use the vertex form of a quadratic function to find the x-intercepts of the graph. To do this, set the function equal to zero and solve for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>.</p> <h2><strong>Q: How do I use the vertex form of a quadratic function to find the maximum or minimum value of the function?</strong></h2> <hr> <p>A: To find the maximum or minimum value of a quadratic function, you can use the vertex form of the function. Since the vertex is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>, you know that the maximum or minimum value of the function is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>.</p> <h2><strong>Q: Can I use the vertex form of a quadratic function to find the equation of the axis of symmetry?</strong></h2> <hr> <p>A: Yes, you can use the vertex form of a quadratic function to find the equation of the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola, and its equation is given by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">x = h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>.</p> <h2><strong>Q: How do I use the vertex form of a quadratic function to find the equation of the parabola?</strong></h2> <hr> <p>A: To find the equation of the parabola, you can use the vertex form of the function. Since the vertex is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>, you know that the equation of the parabola is given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>h</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">f(x) = a(x - h)^2 + k </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></span></p> <h2><strong>Example Problems</strong></h2> <hr> <h3><strong>Problem 1</strong></h3> <p>Find the value of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> in the vertex form of the quadratic function:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">f(x) = a(x - 2)^2 + 3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p> <p>Given point: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>15</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(4, 15)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">15</span><span class="mclose">)</span></span></span></span></p> <h3><strong>Solution</strong></h3> <p>Substituting the point <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>4</mn><mo separator="true">,</mo><mn>15</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(4, 15)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">15</span><span class="mclose">)</span></span></span></span> into the vertex form of the function, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>15</mn><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mn>4</mn><mo>−</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">15 = a(4 - 2)^2 + 3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">15</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p> <p>Simplifying the equation, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>15</mn><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mn>2</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">15 = a(2)^2 + 3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">15</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>15</mn><mo>=</mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">15 = 4a + 3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">15</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p> <p>Subtracting 3 from both sides, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>12</mn><mo>=</mo><mn>4</mn><mi>a</mi></mrow><annotation encoding="application/x-tex">12 = 4a </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">12</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span><span class="mord mathnormal">a</span></span></span></span></span></p> <p>Dividing both sides by 4, we get:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">a = 3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p> <h3><strong>Problem 2</strong></h3> <p>Find the equation of the axis of symmetry of the parabola given by:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f(x) = 2(x - 3)^2 + 2 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">3</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p> <h3><strong>Solution</strong></h3> <p>The equation of the axis of symmetry is given by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>h</mi></mrow><annotation encoding="application/x-tex">x = h</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">h</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> is the vertex of the parabola. In this case, the vertex is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(3, 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>, so the equation of the axis of symmetry is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">x = 3 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p> <h3><strong>Problem 3</strong></h3> <p>Find the maximum value of the quadratic function:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">f(x) = 2(x - 1)^2 + 4 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span></p> <h3><strong>Solution</strong></h3> <p>The maximum value of the function is given by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(h, k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> is the vertex of the parabola. In this case, the vertex is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 4)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">4</span><span class="mclose">)</span></span></span></span>, so the maximum value of the function is:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>k</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">k = 4 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span></p>