Rewriting Expressions in Vertex Form

9.10.4 Rewriting Expressions in Vertex Form

9.10.4 • Rewriting Expressions in Vertex Form

Activity

1. Here is one way to rewrite $3 x^{2} + 12 x + 9$ in vertex form (of a quadratic expression).

a. Study the steps and write a brief explanation of what is happening at each step.

Original expression $3 x^{2} + 12 x + 9$

Step 1 - $3 (x^{2} + 4 x + 3)$

Step 2 - $3 (x^{2} + 4 x + 4 + 3 - 4)$

Step 3 - $3 (x^{2} + 4 x + 4 - 1)$

Step 4 - $3 ((x + 2)^{2} - 1)$

Step 5 - $3 (x + 2)^{2} - 3$

Compare your answers:

Original expression $3 x^{2} + 12 x + 9$

Step 1 - Rewrite the expression with 3 as a factor. $3 (x^{2} + 4 x + 3)$

Step 2 - Add 4 to complete the square. Subtract 4 to keep the value of the expression the same. $3 (x^{2} + 4 x + 4 + 3 - 4)$

Step 3 - Combine like terms so we can see the perfect square, $x^{2} + 4 x + 4$ . $3 (x^{2} + 4 x + 4 - 1)$

Step 4 - Rewrite $x^{2} + 4 x + 4$ as a squared expression. $3 ((x + 2)^{2} - 1)$

Step 5 - Apply the distributive property to put the expression in vertex form. $3 (x + 2)^{2} - 3$

b. What is the $x$ -coordinate of the vertex (of a graph) that represents this expression?

The $x$ -coordinate of the vertex is -2.

c. What is the $y$ -coordinate of the vertex of the graph that represents this expression?

The $y$ -coordinate of the vertex is -3.

d. Does the graph open upward or downward? Explain how you know.

Compare your answer:

The graph opens upward. For example, in an earlier unit, we saw that when the coefficient of the squared term $(x - h)^{2}$ is positive, the graph opens upward.

2. Rewrite $- 2 x^{2} - 4 x + 6$ in vertex form.

Compare your answer:

$- 2 (x + 1)^{2} + 8$

3. Rewrite $4 x^{2} + 24 x + 20$ in vertex form.

Compare your answer:

$4 (x + 3)^{2} - 16$

4. Rewrite $- x^{2} + 20 x$ in vertex form.

Compare your answer:

$- (x - 10)^{2} + 100$

Are you ready for more?

Extending Your Thinking

1. Fill in the missing parts of the function $f (x) = 2 (x - 3) (x - 9)$ in vertex form $f (x) = \underset{―}{a .} (x \underset{―}{b .} \underset{―}{c .})^{2} \underset{―}{d .} \underset{―}{e .}$ without completing the square. (Hint: Think about finding the zeros of the function.)

a. What is the value of a?

b. What is the sign of b?

Subtraction(-)

Addition(+)

Minus (-)

c. What is the value of c?

d. What is the sign of d?

Subtraction(-)

Addition(+)

Minus (-)

e. What is the value of e?

f. Be prepared to show your reasoning for your answers.

Compare your answer:

The zeros of $f$ are 3 and 9. Because the graph of a quadratic function has a vertical line of symmetry across the vertex, the vertex of the graph must have $x$ -coordinate 6. The $y$ -coordinate of the vertex is -18 because $f (6) = - 18$ . This means $f (x)$ can be expressed as $a (x - 6)^{2} - 18$ . If we expand $2 (x - 3) (x - 9)$ , we can see that the coefficient of $x^{2}$ is 2. This means that $a$ in the vertex form has to be 2 as well.

2. Fill in the missing parts of the function $g (x) = 2 (x - 3) (x - 9) + 21$ in vertex form $g (x) = \underset{―}{a .} (x \underset{―}{b .} \underset{―}{c .})^{2} \underset{―}{d .} \underset{―}{e .}$ without completing the square.

a. What is the value of a.

b. What is the sign of b.

Subtraction(-)

Addition(+)

Minus (-)

c. What is the value of c.

d. What is the sign of d.

Subtraction(-)

Addition(+)

Plus (+)

e. What is the value of e.

f. Be prepared to show your reasoning for your answers.

Compare your answer:

Although 3 and 9 are not zeros of $g$ , when $x$ is 3 and when $x$ is 9, $g (x)$ is 21, or $g (3) = g (9) = 21$ . Because the graphs of quadratic functions are symmetric, the $x$ -coordinate of the vertex is still 6. The $y$ -coordinate of the vertex is 3 because $g (6) = 3$ . The value of $a$ is 2 for the same reason as in function $f$ .

Video: Rewriting Expressions in Vertex Form

Watch the following video to learn more about rewriting expressions in vertex form.

Access multimedia content

Self Check

Write

2x^2+20x+20

in vertex form.

$2(x+5)^2-15$
$2(x^2+10x+10)$
$2(x+5)^2-30$
$2(x+5)^2$

Additional Resources

Vertex Form with a Leading Coefficient Besides 1

Write $2 x^{2} + 8 x + 4$ in vertex form (of a quadratic expression).

Original expression $2 x^{2} + 8 x + 4$

Step 1 - Rewrite the expression with 2 as a factor. $2 (x^{2} + 4 x + 2)$

Step 2 - Add 4 to complete the square. Subtract 4 to keep the value of the expression the same. $2 (x^{2} + 4 x + 4 + 2 - 4)$

Step 3 - Combine like terms so we can see the perfect square, $x^{2} + 4 x + 4$ . $2 (x^{2} + 4 x + 4 - 2)$

Step 4 - Rewrite $x^{2} + 4 x + 4$ as a squared expression. $2 ((x + 2)^{2} - 2)$

Step 5 - Apply the distributive property to put the expression in vertex form. $2 (x + 2)^{2} - 4$

Try it

Vertex Form with a Leading Coefficient Besides 1

Write $4 x^{2} + 24 x + 12$ in vertex form.

Here is how to write the quadratic expression in vertex form:

Original expression $4 x^{2} + 24 x + 12$

Step 1 - Rewrite the expression with 4 as a factor. $4 (x^{2} + 6 x + 3)$

Step 2 - Add 9 to complete the square. Subtract 9 to keep the value of the expression the same. $4 (x^{2} + 6 x + 9 + 3 - 9)$

Step 3 - Combine like terms so we can see the perfect square, $x^{2} + 6 x + 9$ . $4 (x^{2} + 6 x + 9 - 6)$

Step 4 - Rewrite $x^{2} + 6 x + 9$ as a squared expression. $4 ((x + 3)^{2} - 6)$

Step 5 - Apply the distributive property to put the expression in vertex form. $4 (x + 3)^{2} - 24$