Convert The Function Y = 2 X 2 − 24 X + 60 Y=2x^2-24x+60 Y = 2 X 2 − 24 X + 60 Into Vertex Form.A) Y = ( 2 X − 15 ) 2 − 2 Y=(2x-15)^2-2 Y = ( 2 X − 15 ) 2 − 2 B) Y = 2 ( X − 6 ) 2 − 8 Y=2(x-6)^2-8 Y = 2 ( X − 6 ) 2 − 8 C) Y = 2 ( X − 6 ) 2 − 12 Y=2(x-6)^2-12 Y = 2 ( X − 6 ) 2 − 12

Introduction

In mathematics, the vertex form of a quadratic function is a way to express the function in the form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. This form is useful for identifying the vertex and the direction of the parabola. In this article, we will learn how to convert the function $y=2x^2-24x+60$ into vertex form.

Understanding the Standard Form

Before we can convert the function into vertex form, we need to understand the standard form of a quadratic function, which is $y=ax^2+bx+c$. In this form, $a$, $b$, and $c$ are constants, and $x$ is the variable. The standard form can be rewritten as $y=a(x-h)^2+k$, where $h=-\frac{b}{2a}$ and $k=c-\frac{b^2}{4a}$.

Converting the Function into Vertex Form

To convert the function $y=2x^2-24x+60$ into vertex form, we need to follow these steps:

1. Identify the values of $a$, $b$, and $c$: In the given function, $a=2$, $b=-24$, and $c=60$.
2. Calculate the value of $h$: Using the formula $h=-\frac{b}{2a}$, we can calculate the value of $h$ as follows:

$$h=-\frac{-24}{2(2)}=\frac{24}{4}=6$$

3. Calculate the value of $k$: Using the formula $k=c-\frac{b^2}{4a}$, we can calculate the value of $k$ as follows:

$$k=60-\frac{(-24)^2}{4(2)}=60-\frac{576}{8}=60-72=-12$$

4. Write the function in vertex form: Now that we have the values of $h$ and $k$, we can write the function in vertex form as follows:

$$y=2(x-6)^2-12$$

Comparing the Options

Now that we have converted the function into vertex form, let's compare it with the given options:

• A) $y=(2x-15)^2-2$
• B) $y=2(x-6)^2-8$
• C) $y=2(x-6)^2-12$

From the options, we can see that option C is the correct answer, as it matches the function we converted into vertex form.

Conclusion

In this article, we learned how to convert the function $y=2x^2-24x+60$ into vertex form. We identified the values of $a$, $b$, and $c$, calculated the value of $h$, and then calculated the value of $k$. Finally, we wrote the function in vertex form and compared it with the given options. We found that option C is the correct answer.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is a way to express the function in the form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. This form is useful for identifying the vertex and the direction of the parabola.

Example Problems

Here are some example problems to help you practice converting a quadratic function into vertex form:

• Convert the function $y=x^2+4x+3$ into vertex form.
• Convert the function $y=2x^2-8x+5$ into vertex form.
• Convert the function $y=x^2-6x+2$ into vertex form.

Solutions

Here are the solutions to the example problems:

• $y=(x+2)^2-1$
• $y=2(x-2)^2-3$
• $y=(x-3)^2-1$

Tips and Tricks

Here are some tips and tricks to help you convert a quadratic function into vertex form:

• Make sure to identify the values of $a$, $b$, and $c$ correctly.
• Use the formula $h=-\frac{b}{2a}$ to calculate the value of $h$.
• Use the formula $k=c-\frac{b^2}{4a}$ to calculate the value of $k$.
• Write the function in vertex form using the values of $h$ and $k$.

Conclusion

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Introduction

In our previous article, we learned how to convert a quadratic function into vertex form. In this article, we will provide a Q&A section to help you understand the concept better.

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

A: The vertex form of a quadratic function is a way to express the function in the form $y=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola.

Q: How do I convert a quadratic function into vertex form?

A: To convert a quadratic function into vertex form, you need to follow these steps:

1. Identify the values of $a$, $b$, and $c$.
2. Calculate the value of $h$ using the formula $h=-\frac{b}{2a}$.
3. Calculate the value of $k$ using the formula $k=c-\frac{b^2}{4a}$.
4. Write the function in vertex form using the values of $h$ and $k$.

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

A: The vertex form of a quadratic function is useful for identifying the vertex and the direction of the parabola. It also helps in graphing the function and finding the maximum or minimum value of the function.

Q: How do I identify the values of $a$, $b$, and $c$ in a quadratic function?

A: To identify the values of $a$, $b$, and $c$ in a quadratic function, you need to look at the coefficients of the terms in the function. The coefficient of the $x^2$ term is $a$, the coefficient of the $x$ term is $b$, and the constant term is $c$.

Q: What is the formula for calculating the value of $h$?

A: The formula for calculating the value of $h$ is $h=-\frac{b}{2a}$.

Q: What is the formula for calculating the value of $k$?

A: The formula for calculating the value of $k$ is $k=c-\frac{b^2}{4a}$.

Q: How do I write a quadratic function in vertex form?

A: To write a quadratic function in vertex form, you need to use the values of $h$ and $k$ in the formula $y=a(x-h)^2+k$.

Q: What are some common mistakes to avoid when converting a quadratic function into vertex form?

A: Some common mistakes to avoid when converting a quadratic function into vertex form include:

• Not identifying the values of $a$, $b$, and $c$ correctly.
• Not using the correct formula for calculating the value of $h$.
• Not using the correct formula for calculating the value of $k$.
• Not writing the function in vertex form correctly.

Q: How can I practice converting quadratic functions into vertex form?

A: You can practice converting quadratic functions into vertex form by:

• Working on example problems.
• Using online resources and worksheets.
• Asking your teacher or tutor for help.
• Practicing with different types of quadratic functions.

Conclusion

In this article, we provided a Q&A section to help you understand the concept of converting a quadratic function into vertex form. We covered topics such as the significance of the vertex form, how to identify the values of $a$, $b$, and $c$, and how to write a quadratic function in vertex form. We also provided some common mistakes to avoid and ways to practice converting quadratic functions into vertex form.