Graphing Quadratic Equations

Using Vertex form

Vertex Form of a Quadratic Function:A quadratic function with vertex at (h, k) can be written as:

2( )y a x h k

The “a” in the formula has the same properties as the “a” in the formula for standard form.

• (a is positive), then the U-shape opens up.• (a is negative), then the -shape opens down.

• “big” |a| narrow graph• “small” |a| wide graph

2( )y x h ka

0a 0a

Graphing Quadratic Equations• Using Vertex form

( , )h kVertex:

To graph: • Plot the vertex (h, k) {be careful of the sign of the h-value}• Use the value of “a” to determine whether the graph opens

up or down. {positive “a” up, negative “a” down}• Use the value of “a” to determine the width of the graph.• Think of “a” as the “initial slope” of the curve.• If a = 1, then move from the vertex up 1 and right 1

to find another point on the graph. • If a = 2, then move from the vertex up 2 and right 1

to find another point on the graph.• If a = -2, then move from the vertex down 2 and

right 1 to find another point on the graph• Etc.

• Use the line of symmetry as a guide to draw a smooth U-shaped curve.

Graphing Quadratic Equations• Using Vertex form

Graphing Quadratic Equations• Using Vertex form

Example: Sketch the graph of 2)1( 3 2y x 2)1( 3 2y x 2)1( 3 2y x

3

2

h

k

1a

up 1 and right 1 from the vertex

Positive a U-shape opens up

Line of

Symmetry

( , ) (3,2)h k Vertex

Graphing Quadratic Equations• Using Vertex form

Example: Sketch the graph of 22( 5) 2y x 22( ( 5)) 2y x 2( ( 52 )) 2y x

5

2

h

k

2a

down 2 and right 1 from the vertex

negative a U-shape opens down

Line of

Symmetry

( , ) ( 5, 2)h k Vertex

Graphing Quadratic Equations• Using Vertex form