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Algebra 1

9.9.2 Different Forms of Quadratics

Algebra 19.9.2 Different Forms of Quadratics

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Activity

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

1.

y = x 2 4 x + 7 y = x 2 4 x + 7

2.

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

3.

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

4.

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

5.

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

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

6.

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

7.

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

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?

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

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.

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