Graphing Graphing Quadratic Quadratic FunctionsFunctions

2 Forms of Quadratic Equations

y = ax2 + bx + c

y = a(x – h)2 + kStandard

FormVertex Form

The axis of symmetry for the parabola is the vertical line through the vertex.

Graphing Using Vertex Form

y = a(x – h)2 + k

Vertex: (h, k)

Axis of symmetry: x = h

VERTICAL LINEIf a is positive, then it opens up.

If a is negative, then it opens down.

Graphing Using Vertex Form

1.Find and sketch the axis of symmetry (opposite of h).

2.Find and plot your vertex (opposite of h, same as k).

3.Construct a table of values to find 2 points on one side of the axis of symmetry (choose 2 x-values above your symmetry value)

Graphing Using Vertex Form

4. Use Symmetry to plot the points on the opposite side of your axis of symmetry.

5. Connect them with a U-shaped curve

Tell whether it opens up or down, axis of symmetry, and name the vertex.

f(x)= -3(x – 2)2 + 5

Vertex: (2, 5)

Axis of symmetry: x = 2

Opens DOWN

a = -3

h = 2 k = 5

y = a(x – h)2 + k.

f(x) = (x + 4)2 – 6a = 1 h = -4 k = -6

Tell whether it opens up or down, axis of symmetry, and name the vertex.

Vertex: (-4, -6)

Axis of symmetry: x = -4Opens UP

You try…

x 2(x + 5)2 - 4

y (x, y)

2f 2( 5) 4x x

Graph

x y (x, y)

21f ( 3) 1

2x x

Graph

21( 3) 1

2x

Graphing Using Standard Form

*Once it is in standard form:1.Find and sketch the axis of symmetry using 2.Find your vertex by substituting your axis of symmetry back into the original equation and solve for y.

a

bx

2

Graphing Using Standard Form

4. Construct a table of values to find 2 points on one side of the axis of symmetry (choose 2 x-values above your symmetry value)

5. Use Symmetry to plot the points on the opposite side of your axis of symmetry.

6. Connect them with a U-shaped curve

*Remember: If a is positive, it opens up, if a is negative, it opens down.

x (x)2 + 8x + 13

y (x, y)

2f 8 13x x x

x b2a

Graph

x -(x)2 + 2x y (x, y)

2f 2x x x

x b2a

Graph

Converting From Converting From Vertex Form to Vertex Form to Standard Form:Standard Form:y = (x – 3)2 + 5

Step 1: FOIL the binomialStep 2: Multiply the “a” term

by what you just foiledStep 3: combine like terms!

Convert the following to Convert the following to standard form:standard form:

y = 2(x – 4)2 + 6

Convert the following to Convert the following to standard form:standard form:

y = (x + 3)² + 4

Step 1: Identify a, b, and cStep 2: find the vertex (h, k)

x-coordinate (h) =

y-coordinate (k) = substitute the value you found for the x coordinate.

Step 3: Substitute a, h, and k into vertex form!

2

ba

Converting From Converting From Standard Form to Standard Form to

Vertex FormVertex Form

Convert the following to vertex form:Convert the following to vertex form:

What is the vertex form of a What is the vertex form of a parabola whose standard parabola whose standard form equation is:form equation is:

Convert the following to vertex form:Convert the following to vertex form: