The Following Equation Of A Quadratic Function Is Given In Standard Form:${ F(x) = 2x^2 + 16x + 35 } F I N D T H E S A M E E Q U A T I O N I N V E R T E X F O R M , T H E N E N T E R I T B E L O W . R O U N D Y O U R A N S W E R S T O T H E N E A R E S T T E N T H I F N E C E S S A R Y . Find The Same Equation In Vertex Form, Then Enter It Below. Round Your Answers To The Nearest Tenth If Necessary. F In D T H Es Am Ee Q U A T I O Nin V Er T E X F Or M , T H E N E N T Er I T B E L O W . R O U N D Yo U R An S W Ers T O T H E N E A Res Tt E N T Hi F N Ecess A Ry . [ F(x) = \square(x -

Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The standard form of a quadratic function is given by the equation $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. However, there is another form of a quadratic function called the vertex form, which is given by the equation $f(x) = a(x - h)^2 + k$. In this article, we will learn how to convert the standard form of a quadratic function to its vertex form.

The Standard Form of a Quadratic Function

The standard form of a quadratic function is given by the equation $f(x) = ax^2 + bx + c$. This form is also known as the general form of a quadratic function. The coefficients $a$, $b$, and $c$ can be any real numbers, and the variable $x$ can be any real number.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by the equation $f(x) = a(x - h)^2 + k$. This form is also known as the vertex form of a quadratic function. The vertex form is a more convenient form to work with, especially when graphing quadratic functions.

Converting the Standard Form to Vertex Form

To convert the standard form of a quadratic function to its vertex form, we need to complete the square. The process of completing the square involves rewriting the quadratic expression in a perfect square trinomial form.

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

The first step in completing the square is to factor out the coefficient of $x^2$. This will make it easier to complete the square.

f(x) = 2x^2 + 16x + 35

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

The next step is to add and subtract the square of half the coefficient of $x$. This will help us to create a perfect square trinomial.

f(x) = 2(x^2 + 8x) + 35

Step 3: Add and Subtract the Square of Half the Coefficient of $x$

Now, we need to add and subtract the square of half the coefficient of $x$. Half of the coefficient of $x$ is 4, and the square of 4 is 16.

f(x) = 2(x^2 + 8x + 16) + 35 - 32

Step 4: Simplify the Expression

Now, we can simplify the expression by combining like terms.

f(x) = 2(x + 4)^2 + 3

Step 5: Write the Equation in Vertex Form

Now that we have completed the square, we can write the equation in vertex form.

f(x) = 2(x - (-4))^2 + 3

Step 6: Simplify the Equation

Finally, we can simplify the equation by combining like terms.

f(x) = 2(x + 4)^2 + 3

Conclusion

In this article, we learned how to convert the standard form of a quadratic function to its vertex form. We used the process of completing the square to rewrite the quadratic expression in a perfect square trinomial form. We also learned how to simplify the expression and write the equation in vertex form. The vertex form of a quadratic function is a more convenient form to work with, especially when graphing quadratic functions.

Example

Let's consider an example to illustrate the process of converting the standard form to vertex form.

Suppose we have the quadratic function $f(x) = 3x^2 + 12x + 15$. We can convert this function to vertex form by completing the square.

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

The first step is to factor out the coefficient of $x^2$.

f(x) = 3(x^2 + 4x) + 15

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

The next step is to add and subtract the square of half the coefficient of $x$. Half of the coefficient of $x$ is 2, and the square of 2 is 4.

f(x) = 3(x^2 + 4x + 4) + 15 - 12

Step 3: Simplify the Expression

Now, we can simplify the expression by combining like terms.

f(x) = 3(x + 2)^2 + 3

Step 4: Write the Equation in Vertex Form

Now that we have completed the square, we can write the equation in vertex form.

f(x) = 3(x - (-2))^2 + 3

Step 5: Simplify the Equation

Finally, we can simplify the equation by combining like terms.

f(x) = 3(x + 2)^2 + 3

Applications of the Vertex Form

The vertex form of a quadratic function has many applications in mathematics and science. Some of the applications include:

• Graphing Quadratic Functions: The vertex form is a more convenient form to work with when graphing quadratic functions.
• Finding the Vertex: The vertex form makes it easy to find the vertex of a quadratic function.
• Solving Quadratic Equations: The vertex form can be used to solve quadratic equations.
• Modeling Real-World Problems: The vertex form can be used to model real-world problems, such as the motion of an object under the influence of gravity.

Conclusion

Introduction

In our previous article, we learned how to convert the standard form of a quadratic function to its vertex form. We also learned about the applications of the vertex form in mathematics and science. In this article, we will answer some frequently asked questions about the vertex form of a quadratic function.

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

A: The vertex form of a quadratic function is given by the equation $f(x) = a(x - h)^2 + k$, where $a$, $h$, and $k$ are constants.

Q: How do I convert the standard form of a quadratic function to its vertex form?

A: To convert the standard form of a quadratic function to its vertex form, you need to complete the square. This involves rewriting the quadratic expression in a perfect square trinomial form.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic expression in a perfect square trinomial form. This involves adding and subtracting the square of half the coefficient of $x$.

Q: How do I find the vertex of a quadratic function in vertex form?

A: To find the vertex of a quadratic function in vertex form, you need to identify the values of $h$ and $k$. The vertex is given by the point $(h, k)$.

Q: What are the applications of the vertex form of a quadratic function?

A: The vertex form of a quadratic function has many applications in mathematics and science, including graphing quadratic functions, finding the vertex, solving quadratic equations, and modeling real-world problems.

Q: Can I use the vertex form to solve quadratic equations?

A: Yes, you can use the vertex form to solve quadratic equations. The vertex form makes it easy to find the solutions to a quadratic equation.

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

A: To graph a quadratic function in vertex form, you need to identify the values of $a$, $h$, and $k$. The graph of the function is a parabola that opens upwards or downwards, depending on the value of $a$.

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 makes it easy to find the vertex, solve quadratic equations, and graph quadratic functions.

Q: Can I use the vertex form to model real-world problems?

A: Yes, you can use the vertex form to model real-world problems, such as the motion of an object under the influence of gravity.

Q: How do I determine the value of $a$ in the vertex form of a quadratic function?

A: To determine the value of $a$ in the vertex form of a quadratic function, you need to look at the coefficient of the $x^2$ term. The value of $a$ is the coefficient of the $x^2$ term.

Q: Can I use the vertex form to find the maximum or minimum value of a quadratic function?

A: Yes, you can use the vertex form to find the maximum or minimum value of a quadratic function. The vertex form makes it easy to find the maximum or minimum value of a quadratic function.

Conclusion

In conclusion, the vertex form of a quadratic function is a more convenient form to work with, especially when graphing quadratic functions, finding the vertex, solving quadratic equations, and modeling real-world problems. We answered some frequently asked questions about the vertex form of a quadratic function and provided examples to illustrate the concepts.

Example Problems

Here are some example problems to help you practice working with the vertex form of a quadratic function:

1. Convert the standard form of the quadratic function $f(x) = 2x^2 + 12x + 15$ to its vertex form.
2. Find the vertex of the quadratic function $f(x) = 3(x - 2)^2 + 4$.
3. Solve the quadratic equation $f(x) = 2(x - 1)^2 + 3 = 0$.
4. Graph the quadratic function $f(x) = 2(x + 2)^2 - 3$.
5. Model the motion of an object under the influence of gravity using the vertex form of a quadratic function.

Answer Key

Here are the answers to the example problems:

1. $f(x) = 2(x + 3)^2 - 3$
2. $(2, 4)$
3. $x = 1$
4. A parabola that opens upwards
5. $f(x) = -\frac{1}{2}x^2 + 2x + 1$

Conclusion

In conclusion, the vertex form of a quadratic function is a powerful tool for graphing quadratic functions, finding the vertex, solving quadratic equations, and modeling real-world problems. We provided examples to illustrate the concepts and answered some frequently asked questions about the vertex form of a quadratic function.