Lesson Objectives

- Learn how to find the vertex form of a parabola

## How to Find the Vertex Form of a Parabola

In this lesson, we want to learn about the vertex form of a parabola. At this point, we should understand that a parabola is the graph of a quadratic function. Additionally, the vertex of a parabola is the lowest point for an upward-facing parabola or the highest point for a downward-facing parabola. In most books, we will see the standard form of a parabola given as: $$f(x)=ax^2 + bx + c$$ Here, we will learn how to put this function in vertex form: $$f(x)=a(x - h)^2 + k$$ This form is helpful, as the vertex is given as: $$(h, k)$$ Let's look at an example.

Example #1: Find the vertex form. $$4x^2 + 56x + 197$$ We can do this by completing the square or by using the vertex formula.

Vertex Formula: $$h=-\frac{b}{2a}$$ $$k=f\left(h\right)$$ Recall that a is the coefficient of the squared variable and b is the coefficient of the variable raised to the first power.

$$a=4$$ $$b=56$$ To find h, let's plug into the formula: $$h=-\frac{b}{2a}$$ $$h=-\frac{56}{8}$$ $$h=-7$$ To find k, we find f(-7): $$4(-7)^2 + 56(-7) + 197=1$$ Now, we fill in the vertex form: $$f(x)=a(x - h)^2 + k$$ $$a=4, h=-7, k=1$$ $$f(x)=4(x + 7)^2 + 1$$

Example #1: Find the vertex form. $$4x^2 + 56x + 197$$ We can do this by completing the square or by using the vertex formula.

Vertex Formula: $$h=-\frac{b}{2a}$$ $$k=f\left(h\right)$$ Recall that a is the coefficient of the squared variable and b is the coefficient of the variable raised to the first power.

$$a=4$$ $$b=56$$ To find h, let's plug into the formula: $$h=-\frac{b}{2a}$$ $$h=-\frac{56}{8}$$ $$h=-7$$ To find k, we find f(-7): $$4(-7)^2 + 56(-7) + 197=1$$ Now, we fill in the vertex form: $$f(x)=a(x - h)^2 + k$$ $$a=4, h=-7, k=1$$ $$f(x)=4(x + 7)^2 + 1$$

#### Skills Check:

Example #1

Write in vertex form. $$f(x)=-2x^2 + 36x - 157$$

Please choose the best answer.

A

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

B

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

C

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

D

$$f(x)=(x - 7)^2 + 9$$

E

$$f(x)=2(x - 4)^2 + 1$$

Example #2

Write in vertex form. $$f(x)=-7x^2 - 56x - 110$$

Please choose the best answer.

A

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

B

$$f(x)=(x + 5)^2 + 1$$

C

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

D

$$f(x)=-7(x + 1)^2 + 6$$

E

$$f(x)=(x - 1)^2 + 3$$

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