1. 4.2 Graphing Quadratic Functions
in Vertex or Intercept Form
• Definitions
• 3 Forms
• Steps for graphing each form
• Examples
• Changing between eqn. forms

2. Quadratic Function
•
A function of the form y=ax2+bx+c
where a≠0 making a u-shaped graph
called a parabola.
Example quadratic equation:

3. Vertex• The lowest or highest point
of a parabola.
Vertex
Axis of symmetry• The vertical line through the vertex of the
parabola.
Axis of
Symmetry

4. Vertex Form Equation
•
•
•
•
y=a(x-h)2+k
If a is positive, parabola opens up
If a is negative, parabola opens down.
The vertex is the point (h,k).
The axis of symmetry is the vertical line
x=h.
Don’t forget about 2 points on either side
of the vertex! (5 points total!)

5. Vertex Form
Each function we just looked at can be written in the
form (x – h)2 + k, where (h , k) is the vertex of the
parabola, and x = h is its axis of symmetry.
(x – h) 2 + k – vertex form
Equation
Vertex
Axis of Symmetry
y = x2 or
y = (x – 0)2 + 0
( 0 , 0)
x=0
y = x2 + 2 or
y = (x – 0)2 + 2
( 0 , 2)
x=0
y = (x – 3)2 or
y = (x – 3)2 + 0
( 3 , 0)
x=3

6. Example 1: Graph y = (x + 2)2 + 1
Analyze y = (x + 2)2 + 1.
•
• Step 1 Plot the vertex (-2 , 1)
• Step 2 Draw the axis of symmetry, x = -2.
• Step 3 Find and plot two points on one side
, such as (-1, 2) and (0 , 5).
• Step 4 Use symmetry to complete the graph,
or find two points on the
• left side of the vertex.

7. Your Turn!
• Analyze and Graph:
y = (x + 4)2 - 3.
(-4,-3)

8. Example 2: Graph
y= -.5(x+3)2+4
•
•
•
•
a is negative (a = -.5), so parabola opens down.
Vertex is (h,k) or (-3,4)
Axis of symmetry is the vertical line x = -3
Table of values
x y
-1 2
Vertex (-3,4)
-2 3.5
(-4,3.5)
(-2,3.5)
-3 4
-4 3.5
(-5,2)
(-1,2)
-5 2
x=-3

9. Now you try one!
y=2(x-1)2+3
• Open up or down?
• Vertex?
• Axis of symmetry?
• Table of values with 4 points (other
than the vertex?

10. (-1, 11)
(3,11)
X=1
(0,5)
(2,5)
(1,3)

11. Intercept Form Equation
y=a(x-p)(x-q)
•
•
•
•
•
The x-intercepts are the points (p,0) and (q,0).
+
The axis of symmetry is the vertical line x= p 2 q
p +q
The x-coordinate of the vertex is 2
To find the y-coordinate of the vertex, plug the xcoord. into the equation and solve for y.
If a is positive, parabola opens up
If a is negative, parabola opens down.

12. Example 3: Graph y=-(x+2)(x-4)
• Since a is negative,
•
•
parabola opens down.
The x-intercepts are (2,0) and (4,0)
To find the x-coord. of
p+q
the vertex, use
2
x=
•The axis of symmetry
is the vertical line x=1
(from the x-coord. of
the vertex)
(1,9)
−2+4 2
= =1
2
2
• To find the y-coord.,
plug 1 in for x.
y = −(1 + 2)(1 − 4) = −(3)(−3) = 9
(-2,0)
(4,0)
x=1

13. Now you try one!
y=2(x-3)(x+1)
• Open up or down?
• X-intercepts?
• Vertex?
• Axis of symmetry?

14. x=1
(-1,0)
(3,0)
(1,-8)

15. Changing from vertex or intercepts
form to standard form
• The key is to FOIL! (first, outside, inside,
last)
• Ex: y=-(x+4)(x-9)
=-(x2-9x+4x-36)
=-(x2-5x-36)
y=-x2+5x+36
Ex: y=3(x-1)2+8
=3(x-1)(x-1)+8
=3(x2-x-x+1)+8
=3(x2-2x+1)+8
=3x2-6x+3+8
y=3x2-6x+11

16. Challenge Problem
• Write the equation of the graph in vertex form.
y = 3( x + 2) 2 − 4