The Table Of Values Represents A Linear Function.${ \begin{tabular}{||c|c||} \hline X X X & Y Y Y \ \hline -3 & 9 \ \hline 3 & -1 \ \hline 9 & -11 \ \hline \end{tabular} }$Enter An Equation In The Form Y = M X + B Y = Mx + B Y = M X + B That Represents

Introduction

A linear function is a type of function that can be represented in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The table of values represents a linear function, and in this article, we will learn how to enter an equation in the form $y = mx + b$ that represents the given table of values.

Understanding the Table of Values

The table of values is a collection of points that represent the input and output values of a function. In this case, the table of values is given as:

$x$	$y$
-3	9
3	-1
9	-11

The table of values shows that when $x$ is -3, the corresponding value of $y$ is 9. Similarly, when $x$ is 3, the corresponding value of $y$ is -1, and when $x$ is 9, the corresponding value of $y$ is -11.

Finding the Slope

The slope of a linear function is a measure of how much the function changes as the input value changes. In this case, we can find the slope by using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the table of values.

Let's use the points (-3, 9) and (3, -1) to find the slope:

$$m = \frac{-1 - 9}{3 - (-3)} = \frac{-10}{6} = -\frac{5}{3}$$

So, the slope of the linear function is $-\frac{5}{3}$.

Finding the Y-Intercept

The y-intercept of a linear function is the value of $y$ when $x$ is 0. In this case, we can find the y-intercept by using the point (0, b) and the slope $m$.

We know that the point (0, b) lies on the line, so we can substitute $x = 0$ and $y = b$ into the equation $y = mx + b$:

$$b = m(0) + b$$

Simplifying the equation, we get:

$$b = b$$

This equation is true for any value of $b$, so we can use any point on the table of values to find the y-intercept.

Let's use the point (-3, 9) to find the y-intercept:

$$9 = -\frac{5}{3}(-3) + b$$

Simplifying the equation, we get:

$$9 = 5 + b$$

Subtracting 5 from both sides, we get:

$$b = 4$$

So, the y-intercept of the linear function is 4.

Writing the Equation

Now that we have found the slope and y-intercept, we can write the equation of the linear function in the form $y = mx + b$:

$$y = -\frac{5}{3}x + 4$$

This equation represents the table of values and is a linear function.

Conclusion

In this article, we learned how to enter an equation in the form $y = mx + b$ that represents a given table of values. We found the slope and y-intercept of the linear function using the points on the table of values and wrote the equation of the linear function in the form $y = mx + b$. The equation of the linear function is $y = -\frac{5}{3}x + 4$.

Example Problems

1. Find the equation of the linear function represented by the table of values:

$x$	$y$
2	5
4	9
6	13

2. Find the equation of the linear function represented by the table of values:

$x$	$y$
-2	3
0	5
2	7

Solutions

1. To find the equation of the linear function, we need to find the slope and y-intercept. Let's use the points (2, 5) and (4, 9) to find the slope:

$$m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2$$

So, the slope of the linear function is 2.

Now, let's use the point (2, 5) to find the y-intercept:

$$5 = 2(2) + b$$

Simplifying the equation, we get:

$$5 = 4 + b$$

Subtracting 4 from both sides, we get:

$$b = 1$$

So, the y-intercept of the linear function is 1.

The equation of the linear function is:

$$y = 2x + 1$$

2. To find the equation of the linear function, we need to find the slope and y-intercept. Let's use the points (-2, 3) and (0, 5) to find the slope:

$$m = \frac{5 - 3}{0 - (-2)} = \frac{2}{2} = 1$$

So, the slope of the linear function is 1.

Now, let's use the point (-2, 3) to find the y-intercept:

$$3 = 1(-2) + b$$

Simplifying the equation, we get:

$$3 = -2 + b$$

Adding 2 to both sides, we get:

$$b = 5$$

So, the y-intercept of the linear function is 5.

The equation of the linear function is:

y = x + 5$<br/> **Q&A: The Table of Values Represents a Linear Function** =====================================================

Frequently Asked Questions

Q: What is a linear function?

A: A linear function is a type of function that can be represented in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I find the slope of a linear function?

A: To find the slope of a linear function, you can use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1} </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><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_1, y_1)</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"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_2, y_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"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;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 class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> are two points on the table of values.</p> <h2><strong>Q: How do I find the y-intercept of a linear function?</strong></h2> <p>A: To find the y-intercept of a linear function, you can use the point (0, b) and the slope <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</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">m</span></span></span></span>. You can substitute <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x = 0</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">0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y = b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</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">b</span></span></span></span> into the equation <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y = mx + b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</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:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</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:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> and solve for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</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">b</span></span></span></span>.</p> <h2><strong>Q: What is the equation of a linear function?</strong></h2> <p>A: The equation of a linear function is in the form <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y = mx + b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</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:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</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:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</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">m</span></span></span></span> is the slope and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</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">b</span></span></span></span> is the y-intercept.</p> <h2><strong>Q: How do I write the equation of a linear function?</strong></h2> <p>A: To write the equation of a linear function, you need to find the slope and y-intercept. Once you have found the slope and y-intercept, you can write the equation of the linear function in the form <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y = mx + b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</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:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</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:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>.</p> <h2><strong>Q: What is the significance of the slope and y-intercept in a linear function?</strong></h2> <p>A: The slope and y-intercept are important components of a linear function. The slope represents the rate of change of the function, while the y-intercept represents the point where the function intersects the y-axis.</p> <h2><strong>Q: Can a linear function have a negative slope?</strong></h2> <p>A: Yes, a linear function can have a negative slope. A negative slope indicates that the function is decreasing as the input value increases.</p> <h2><strong>Q: Can a linear function have a zero slope?</strong></h2> <p>A: Yes, a linear function can have a zero slope. A zero slope indicates that the function is a horizontal line.</p> <h2><strong>Q: Can a linear function have a fractional slope?</strong></h2> <p>A: Yes, a linear function can have a fractional slope. A fractional slope indicates that the function is a line with a slope that is a fraction.</p> <h2><strong>Q: Can a linear function have a negative y-intercept?</strong></h2> <p>A: Yes, a linear function can have a negative y-intercept. A negative y-intercept indicates that the function intersects the y-axis below the origin.</p> <h2><strong>Q: Can a linear function have a zero y-intercept?</strong></h2> <p>A: Yes, a linear function can have a zero y-intercept. A zero y-intercept indicates that the function intersects the y-axis at the origin.</p> <h2><strong>Q: Can a linear function have a fractional y-intercept?</strong></h2> <p>A: Yes, a linear function can have a fractional y-intercept. A fractional y-intercept indicates that the function intersects the y-axis at a point that is a fraction of the unit.</p> <h2><strong>Conclusion</strong></h2> <p>In this article, we have answered some of the frequently asked questions about the table of values represents a linear function. We have discussed the definition of a linear function, how to find the slope and y-intercept, and how to write the equation of a linear function. We have also discussed the significance of the slope and y-intercept in a linear function and answered some common questions about linear functions.</p>$$