Using the x-intercepts to Rewrite a Quadratic in Graphing Form

Graphing Form for a Parabola

y = a(x – h)2 + k ( h, k ): The Vertex

The value of a

opposite

same

Positive: Opens Up If it Increases: Vertical Stretch

Negative: Opens Down If it Decreases: Vertical Compression

Different Forms For a Quadratic

Parent Graph: y = x2

Factored Form: y = __( __x ± __ )( __x ± __)

Standard form: y = ax2 + bx + c

Graphing Form: y = a(x – h)2 + k

Same “a”

Justification that the “a” in Standard and Graphing Form are the same

4 1 1 7y x x 24 1 7y x

24 1 7y x x x

24 2 1 7y x x 24 8 4 7y x x 24 8 3y x x

Same a!

Standard Form to Graphing Form: Factoring

Use an algebraic method to write in graphing form.

22 1 32y x

20 2 4 30x x 20 2 15x x

0 5 3x x 3 0x

5 3 22 2 1x

2

22 4 30y x x 1. Find the value of a: 2. Find the x-intercepts

5 0x 3x 5x

3. Average the x-intercepts for h

232 1 4 21 30y

4. Substitute h into the rule for k

5. Substitute a, h, k into the graphing form

WARNING: This method does not work if there are no x-intercepts

Instead of Factoring, use the Quadratic Formula

Standard Form to Graphing Form: Quadratic Formula

Use an algebraic method to write in graphing form.

22 1 32y x

20 2 4 30x x 20 2 15x x

22 2 4 1 15

2 1x

5 3 22 2 1x

2

22 4 30y x x 1. Find the value of a: 2. Find the x-intercepts

2 642x

3x 5x

232 1 4 21 30y

WARNING: This method does not work if there are no x-intercepts

3. Average the x-intercepts for h

4. Substitute h into the rule for k

5. Substitute a, h, k into the graphing form