Convert This Quadratic Function To Vertex Form.$\[ Y = X^2 + 14x + 48 \\]Then Determine The Minimum Value Of The Function. Enter Your Answers In The Boxes.Vertex Form: $\[ Y = \boxed{\ } \\]Minimum: $\[ \boxed{\ } \\]

Introduction

In mathematics, quadratic functions are a fundamental concept in algebra and calculus. They are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, and the motion of an object under the influence of a force. One of the most important forms of a quadratic function is the vertex form, which provides valuable information about the function's behavior, such as its minimum or maximum value. In this article, we will learn how to convert a quadratic function from standard form to vertex form and determine the minimum value of the function.

Standard Form to Vertex Form

The standard form of a quadratic function is given by:

${ y = ax^2 + bx + c }$

where $a$, $b$, and $c$ are constants. To convert this function to vertex form, we need to complete the square. The vertex form of a quadratic function is given by:

${ y = a(x - h)^2 + k }$

where $(h, k)$ is the vertex of the parabola.

Step 1: Factor out the Coefficient of $x^2$

The first step in converting the standard form to vertex form is to factor out the coefficient of $x^2$. In this case, the coefficient of $x^2$ is 1, so we can write:

${ y = x^2 + 14x + 48 }$

Step 2: Add and Subtract the Square of Half the Coefficient of $x$

Next, we need to add and subtract the square of half the coefficient of $x$. In this case, the coefficient of $x$ is 14, so we need to add and subtract $(14/2)^2 = 49$:

${ y = x^2 + 14x + 49 - 49 + 48 }$

Step 3: Factor the Perfect Square

Now, we can factor the perfect square:

${ y = (x + 7)^2 - 49 + 48 }$

Step 4: Simplify the Expression

Finally, we can simplify the expression:

${ y = (x + 7)^2 - 1 }$

Vertex Form

The vertex form of the quadratic function is:

${ y = (x + 7)^2 - 1 }$

Minimum Value

To find the minimum value of the function, we need to find the value of $y$ when $x = -7$. Plugging in $x = -7$ into the vertex form, we get:

${ y = (-7 + 7)^2 - 1 }$

${ y = 0^2 - 1 }$

${ y = -1 }$

Therefore, the minimum value of the function is $-1$.

Conclusion

In this article, we learned how to convert a quadratic function from standard form to vertex form and determine the minimum value of the function. We used the method of completing the square to convert the standard form to vertex form and found the minimum value of the function by plugging in the value of $x$ into the vertex form. This technique is useful in various applications, such as modeling real-world phenomena and solving optimization problems.

Example Problems

1. Convert the quadratic function $y = x^2 + 10x + 25$ to vertex form and determine the minimum value of the function.
2. Convert the quadratic function $y = x^2 - 6x + 8$ to vertex form and determine the minimum value of the function.

Answer Key

1. Vertex form: $y = (x + 5)^2 - 25$ Minimum value: $-25$
2. Vertex form: $y = (x - 3)^2 + 2$ Minimum value: $2$

Discussion

• What is the significance of the vertex form of a quadratic function?
• How does the vertex form help in determining the minimum or maximum value of a quadratic function?
• Can you think of any real-world applications of quadratic functions and their vertex form?

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
  Quadratic Functions: Frequently Asked Questions =====================================================

Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, such as physics, engineering, economics, and computer science. In our previous article, we learned how to convert a quadratic function from standard form to vertex form and determine the minimum value of the function. In this article, we will answer some frequently asked questions about quadratic functions.

Q&A

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. It is typically written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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

A: The vertex form of a quadratic function is given by $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful in determining the minimum or maximum value of the function.

Q: How do I convert a quadratic function from standard form to vertex form?

A: To convert a quadratic function from standard form to vertex form, you need to complete the square. This involves adding and subtracting the square of half the coefficient of $x$.

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

A: The vertex form of a quadratic function is significant because it provides valuable information about the function's behavior, such as its minimum or maximum value. It is also useful in solving optimization problems and modeling real-world phenomena.

Q: Can you give an example of a quadratic function in real-world application?

A: Yes, a quadratic function can be used to model the trajectory of a projectile. For example, the height of a projectile as a function of time can be modeled by the quadratic function $y = -4.9t^2 + 20t + 2$, where $y$ is the height in meters and $t$ is the time in seconds.

Q: How do I determine the minimum value of a quadratic function?

A: To determine the minimum value of a quadratic function, you need to find the value of $y$ when $x = -b/2a$. This is because the vertex of the parabola is located at the point $(-b/2a, f(-b/2a))$.

Q: Can you give an example of a quadratic function with a maximum value?

A: Yes, a quadratic function can have a maximum value. For example, the function $y = -x^2 + 4x - 3$ has a maximum value of $5$ when $x = 2$.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the points on the coordinate plane and draw a smooth curve through them. You can also use a graphing calculator or software to graph the function.

Q: Can you give an example of a quadratic function with a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. For example, the function $y = -x^2 + 2x - 3$ has a negative leading coefficient.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions. We covered topics such as the definition of a quadratic function, the vertex form, converting from standard form to vertex form, and determining the minimum value of a quadratic function. We also provided examples of quadratic functions in real-world applications and discussed how to graph a quadratic function.

Example Problems

1. Convert the quadratic function $y = x^2 + 6x + 8$ to vertex form and determine the minimum value of the function.
2. Graph the quadratic function $y = -x^2 + 4x - 3$ and determine its maximum value.
3. Find the vertex of the parabola given by the quadratic function $y = x^2 - 4x + 2$.

Answer Key

1. Vertex form: $y = (x + 3)^2 - 5$ Minimum value: $-5$
2. Graph: The graph of the function is a parabola that opens downward with a maximum value of $5$ at $x = 2$.
3. Vertex: The vertex of the parabola is located at the point $(2, -1)$.

Discussion

• What is the significance of the vertex form of a quadratic function?
• How does the vertex form help in determining the minimum or maximum value of a quadratic function?
• Can you think of any real-world applications of quadratic functions and their vertex form?

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer