The Following Equation Of A Quadratic Function Is Given In Standard Form: F ( X ) = − 2 X 2 + 12 X − 13 F(x) = -2x^2 + 12x - 13 F ( X ) = − 2 X 2 + 12 X − 13 Find The Same Equation In Vertex Form, Then Enter It Below. Round Your Answers To The Nearest Tenth If Necessary.$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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will discuss how to convert a quadratic function from standard form to vertex form.

Standard Form to Vertex Form

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To convert a quadratic function from standard form to vertex form, we need to complete the square.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This can be done by adding and subtracting a constant term to the expression.

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

The first step in completing the square is to factor out the coefficient of $x^2$ from the quadratic expression. In this case, the coefficient of $x^2$ is $-2$, so we can factor it out as follows:

$$f(x) = -2(x^2 - 6x) - 13$$

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 $x$ inside the parentheses. The coefficient of $x$ is $6$, so half of it is $3$. The square of $3$ is $9$, so we can add and subtract $9$ inside the parentheses as follows:

$$f(x) = -2(x^2 - 6x + 9 - 9) - 13$$

Step 3: Simplify the expression

Now we can simplify the expression by combining like terms:

$$f(x) = -2(x^2 - 6x + 9) + 18 - 13$$

$$f(x) = -2(x^2 - 6x + 9) + 5$$

Step 4: Write the expression in vertex form

Finally, we can write the expression in vertex form by factoring the perfect square trinomial:

$$f(x) = -2(x - 3)^2 + 5$$

Conclusion

In this article, we discussed how to convert a quadratic function from standard form to vertex form. We used the technique of completing the square to rewrite the quadratic expression in the form of a perfect square trinomial. The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. We can use this form to find the vertex of a parabola and to graph the parabola.

Vertex Form of the Given Quadratic Function

The vertex form of the given quadratic function is:

$$f(x) = -2(x - 3)^2 + 5$$

Vertex of the Parabola

The vertex of the parabola is given by the point $(h, k)$, where $h$ is the value of $x$ that makes the expression inside the parentheses equal to zero, and $k$ is the value of the expression outside the parentheses. In this case, the vertex is $(3, 5)$.

Graph of the Parabola

The graph of the parabola is a downward-facing parabola with a vertex at $(3, 5)$. The parabola opens downward because the coefficient of the squared term is negative.

Example Problems

Problem 1

Find the vertex form of the quadratic function $f(x) = 3x^2 - 12x + 7$.

Solution

To find the vertex form of the quadratic function, we need to complete the square. We can do this by factoring out the coefficient of $x^2$ and then adding and subtracting the square of half the coefficient of $x$ inside the parentheses.

$$f(x) = 3(x^2 - 4x) + 7$$

$$f(x) = 3(x^2 - 4x + 4 - 4) + 7$$

$$f(x) = 3(x^2 - 4x + 4) - 12 + 7$$

$$f(x) = 3(x - 2)^2 - 5$$

Problem 2

Find the vertex of the parabola given by the quadratic function $f(x) = 2x^2 - 8x + 3$.

Solution

To find the vertex of the parabola, we need to find the values of $h$ and $k$ in the vertex form of the quadratic function. We can do this by factoring out the coefficient of $x^2$ and then adding and subtracting the square of half the coefficient of $x$ inside the parentheses.

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

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

$$f(x) = 2(x^2 - 4x + 4) - 8 + 3$$

$$f(x) = 2(x - 2)^2 - 5$$

The vertex of the parabola is given by the point $(h, k)$, where $h$ is the value of $x$ that makes the expression inside the parentheses equal to zero, and $k$ is the value of the expression outside the parentheses. In this case, the vertex is $(2, -5)$.

Conclusion

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

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

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 factoring out the coefficient of $x^2$ and then adding and subtracting the square of half the coefficient of $x$ inside the parentheses.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This can be done by adding and subtracting a constant term to the expression.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to find the values of $h$ and $k$ in the vertex form of the quadratic function. The vertex is given by the point $(h, k)$, where $h$ is the value of $x$ that makes the expression inside the parentheses equal to zero, and $k$ is the value of the expression outside the parentheses.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the highest or lowest point on the graph of the parabola. It is also the point where the parabola changes direction.

Q: How do I graph a parabola?

A: To graph a parabola, you need to find the vertex of the parabola and then use the vertex form of the quadratic function to determine the direction and shape of the parabola.

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

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

• Not factoring out the coefficient of $x^2$
• Not adding and subtracting the square of half the coefficient of $x$ inside the parentheses
• Not simplifying the expression after completing the square
• Not writing the expression in vertex form

Q: How do I check my work when converting a quadratic function from standard form to vertex form?

A: To check your work when converting a quadratic function from standard form to vertex form, you can:

• Plug in the values of $x$ and $y$ into the original and vertex forms of the quadratic function to see if they are equal
• Graph the original and vertex forms of the quadratic function to see if they are the same
• Use a calculator to graph the quadratic function and check if the vertex is correct

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including:

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a function
• Determining the shape and direction of a parabola
• Solving problems in physics, engineering, and economics

Q: How do I use quadratic functions in real-world applications?

A: To use quadratic functions in real-world applications, you need to:

• Understand the concept of quadratic functions and how to convert them from standard form to vertex form
• Be able to graph quadratic functions and find their vertices
• Be able to use quadratic functions to model real-world problems and solve them

Q: What are some common types of quadratic functions?

A: Some common types of quadratic functions include:

• Linear quadratic functions: $f(x) = ax + b$
• Quadratic functions with a positive leading coefficient: $f(x) = ax^2 + bx + c$
• Quadratic functions with a negative leading coefficient: $f(x) = -ax^2 + bx + c$
• Quadratic functions with a zero leading coefficient: $f(x) = bx + c$

Q: How do I determine the type of quadratic function I am working with?

A: To determine the type of quadratic function you are working with, you need to:

• Look at the leading coefficient of the quadratic function
• Determine if the leading coefficient is positive, negative, or zero
• Use this information to determine the type of quadratic function you are working with.