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Different Forms of Quadratics

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 y = x 2 − 4 x + 7

Compare your answer:

y = ( x − 2 ) 2 + 3 y = ( x − 2 ) 2 + 3

2.

y = ( x − 1 ) ( x + 3 ) y = ( x − 1 ) ( x + 3 )

Compare your answer:

y = ( x + 1 ) 2 − 4 y = ( x + 1 ) 2 − 4

3.

y = ( x − 2 ) ( x + 2 ) y = ( x − 2 ) ( x + 2 )

Compare your answer:

y = ( x − 0 ) 2 − 4 y = ( x − 0 ) 2 − 4

4.

y = − x 2 − 2 x − 6 y = − x 2 − 2 x − 6

Compare your answer:

y = − ( x + 1 ) 2 − 5 y = − ( x + 1 ) 2 − 5

5.

y = 2 x 2 − 12 x + 22 y = 2 x 2 − 12 x + 22

Compare your answer:

y = 2 ( x − 3 ) 2 + 4 y = 2 ( x − 3 ) 2 + 4

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

6.

y = ( x − 2 ) 2 + 7 y = ( x − 2 ) 2 + 7

Compare your answer:

y = x 2 − 4 x + 11 y = x 2 − 4 x + 11

7.

y = ( x + 3 ) 2 − 2 y = ( x + 3 ) 2 − 2

Compare your answer:

y = x 2 + 6 x + 7 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 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 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 H ( t ) = − 16 ( t 2 − 5 t ) + 40

Step 2 - Complete the square.

( − 5 2 ) 2 = 25 4 ( − 5 2 ) 2 = 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 ) ( − 16 ) outside of the parentheses. This makes the constant equation to be ( − 16 × 25 4 = − 100 ( − 16 × 25 4 = − 100 . Because a ( − 100 ) ( − 100 ) was really added in, then a positive 100 will balance the equation.

H ( t ) = − 16 ( t 2 − 5 t + 25 4 ) + 40 − ( − 100 ) H ( t ) = − 16 ( t 2 − 5 t + 25 4 ) + 40 − ( − 100 )

H ( t ) = − 16 ( t 2 − 5 t + 25 4 ) + 40 + 100 H ( t ) = − 16 ( t 2 − 5 t + 25 4 ) + 40 + 100

Step 4 - Factor and simplify.

H ( t ) = − 16 ( t − 5 2 ) 2 + 140 H ( t ) = − 16 ( t − 5 2 ) 2 + 140

Step 5 - Find the vertex.

( 5 2 , 140 ) ( 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?
  1. f ( x ) = x 2 + 4 x + 19
  2. f ( x ) = x 2 − 4 x + 19
  3. f ( x ) = x 2 − 4 x + 11
  4. 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 f ( x ) = a x 2 + b x + c where a a , b b , c c , are real numbers and a ≠ 0 a ≠ 0 .

The vertex form of a quadratic function is f ( x ) = a ( x − h ) 2 + k f ( x ) = a ( x − h ) 2 + k where ( h , k ) ( h , k ) is the vertex.

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

Example 1

Write the function f ( x ) = ( x − 2 ) ( x + 8 ) 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 ) ( x + 8 )

f ( x ) = x 2 − 2 x + 8 x − 16 f ( x ) = x 2 − 2 x + 8 x − 16

Step 2 - Simplify.

f ( x ) = x 2 + 6 x − 16 f ( x ) = x 2 + 6 x − 16

Example 2

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

Hint: Start by using the answer from example 1.

Step 1 - Complete the square.

( b 2 ) 2 = ( 6 2 ) 2 = 9 ( b 2 ) 2 = ( 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 f ( x ) = x 2 + 6 x + 9 − 16 − 9

Finally, factor the trinomial and simplify.
f ( x ) = ( x + 3 ) 2 − 25 f ( x ) = ( x + 3 ) 2 − 25

Example 3

Rewrite the function in standard form:
f ( x ) = 6 ( x − 2 ) 2 + 4 f ( x ) = 6 ( x − 2 ) 2 + 4

Step 1 - Expand the square.

6 ( x − 2 ) 2 + 4 6 ( x − 2 ) 2 + 4 6 ( x 2 − 4 x + 4 ) + 4 6 ( x 2 − 4 x + 4 ) + 4

Step 2 - Distribute the coefficient.

6 x 2 − 24 x + 24 + 4 6 x 2 − 24 x + 24 + 4

Step 3 - Simplify like terms
f ( x ) = 6 x 2 − 24 x + 28 f ( x ) = 6 x 2 − 24 x + 28

Example 4

Rewrite the function in standard form:
f ( x ) = 5 ( x − 3 ) 2 + 2 f ( x ) = 5 ( x − 3 ) 2 + 2

Step 1 - Expand the square.

5 ( x − 3 ) 2 + 2 5 ( x − 3 ) 2 + 2

5 ( x 2 − 6 x + 9 ) + 2 5 ( x 2 − 6 x + 9 ) + 2

Step 2 - Distribute the coefficient.

5 x 2 − 30 x + 45 + 2 5 x 2 − 30 x + 45 + 2

Step 3 - Simplify like terms
f ( x ) = 5 x 2 − 30 x + 47 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 ) 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 + 5 ) ( x + 3 )

f ( x ) = x 2 + 5 x + 3 x + 15 f ( x ) = x 2 + 5 x + 3 x + 15

Then, simplify.

f ( x ) = x 2 + 8 x + 15 f ( x ) = x 2 + 8 x + 15

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

( b 2 ) 2 = ( 8 2 ) 2 = 16 ( b 2 ) 2 = ( 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 f ( x ) = x 2 + 8 x + 16 + 15 − 16

Finally, factor the trinomial and simplify.

f ( x ) = ( x + 4 ) 2 − 1 f ( x ) = ( x + 4 ) 2 − 1