Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

9.9.2 Different Forms of Quadratics

9.9.2 • Different Forms of Quadratics

Activity

For questions 1 – 5, write each function in vertex form.

1.

$y = x^{2} - 4 x + 7$

Compare your answer:

$y = (x - 2)^{2} + 3$

2.

$y = (x - 1) (x + 3)$

Compare your answer:

$y = (x + 1)^{2} - 4$

3.

$y = (x - 2) (x + 2)$

Compare your answer:

$y = (x - 0)^{2} - 4$

4.

$y = - x^{2} - 2 x - 6$

Compare your answer:

$y = - (x + 1)^{2} - 5$

5.

$y = 2 x^{2} - 12 x + 22$

Compare your answer:

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

For questions 6 and 7, write each function in standard form.

6.

$y = (x - 2)^{2} + 7$

Compare your answer:

$y = x^{2} - 4 x + 11$

7.

$y = (x + 3)^{2} - 2$

Compare your answer:

$y = x^{2} + 6 x + 7$

Are you ready for more?

Extending Your Thinking

1.

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball’s height above ground can be modeled by the equation $H (t) = - 16 t^{2} + 80 t + 40$ . What is the maximum height?

Compare your answer:

To find the maximum height, find the vertex. Write the function in vertex form.

$H (t) = - 16 t^{2} + 80 t + 40$

Step 1 - Factor -16 out of the first two terms.

$H (t) = - 16 (t^{2} - 5 t) + 40$

Step 2 - Complete the square.

$(\frac{- 5}{2})^{2} = \frac{25}{4}$

Step 3 - Add the constant to complete the square and subtract from the 40 to balance the equation. Note that because there is a $(- 16)$ outside of the parentheses. This makes the constant equation to be $(- 16 \times \frac{25}{4} = - 100$ . Because a $(- 100)$ was really added in, then a positive 100 will balance the equation.

$H (t) = - 16 (t^{2} - 5 t + \frac{25}{4}) + 40 - (- 100)$

$H (t) = - 16 (t^{2} - 5 t + \frac{25}{4}) + 40 + 100$

Step 4 - Factor and simplify.

$H (t) = - 16 (t - \frac{5}{2})^{2} + 140$

Step 5 - Find the vertex.

$(\frac{5}{2}, 140)$

Therefore, the maximum height is 140 feet. (It reaches this high point after 2.5 seconds.)

Self Check

Which of the following is

f(x)=(x-2)^2+15

written in standard form?

$f(x)=x^2+4x+19$
$f(x)=x^2-4x+19$
$f(x)=x^2-4x+11$
$f(x)=x^2+19$

Additional Resources

Writing Quadratics in Different Forms

Three Common Quadratic Forms:

The standard form of a quadratic function is $f (x) = a x^{2} + b x + c$ where $a$ , $b$ , $c$ , are real numbers and $a \neq 0$ .

The vertex form of a quadratic function is $f (x) = a (x - h)^{2} + k$ where $(h, k)$ is the vertex.

The factored form of the quadratic function is $f (x) = (x - a) (x - b)$ where $a$ and $b$ are roots of $f (x)$ .

Example 1

Write the function $f (x) = (x - 2) (x + 8)$ in standard form.

Step 1 - multiply the binomials using distribution (FOIL).

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

$f (x) = x^{2} - 2 x + 8 x - 16$

Step 2 - Simplify.

$f (x) = x^{2} + 6 x - 16$

Example 2

Write the function $f (x) = (x - 2) (x + 8)$ in vertex form.

Hint: Start by using the answer from example 1.

Step 1 - Complete the square.

$(\frac{b}{2})^{2} = (\frac{6}{2})^{2} = 9$

Step 2 - Balance the equation. Add the constant to the quadratic to complete the square and subtract it from the original constant. $f (x) = x^{2} + 6 x + 9 - 16 - 9$

Finally, factor the trinomial and simplify.
$f (x) = (x + 3)^{2} - 25$

Example 3

Rewrite the function in standard form:
$f (x) = 6 (x - 2)^{2} + 4$

Step 1 - Expand the square.

$6 (x - 2)^{2} + 4$ $6 (x^{2} - 4 x + 4) + 4$

Step 2 - Distribute the coefficient.

$6 x^{2} - 24 x + 24 + 4$

Step 3 - Simplify like terms
$f (x) = 6 x^{2} - 24 x + 28$

Example 4

Rewrite the function in standard form:
$f (x) = 5 (x - 3)^{2} + 2$

Step 1 - Expand the square.

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

$5 (x^{2} - 6 x + 9) + 2$

Step 2 - Distribute the coefficient.

$5 x^{2} - 30 x + 45 + 2$

Step 3 - Simplify like terms
$f (x) = 5 x^{2} - 30 x + 47$

Try it

Writing Quadratic Functions in Equivalent Forms

Write the function $f (x) = (x + 5) (x + 3)$ in vertex form.

Compare your answer:

First, multiply the binomials using distribution (FOIL).

$f (x) = (x + 5) (x + 3)$

$f (x) = x^{2} + 5 x + 3 x + 15$

Then, simplify.

$f (x) = x^{2} + 8 x + 15$

Now, write this function in vertex form. Begin by completing the square.

$(\frac{b}{2})^{2} = (\frac{8}{2})^{2} = 16$

Balance the equation. Add the constant to the quadratic to complete the square and subtract it from the original constant.

$f (x) = x^{2} + 8 x + 16 + 15 - 16$

Finally, factor the trinomial and simplify.

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