Daily Check
1. Factor: 3x2 + 10x + 8
2. Factor and Solve: 2x2 - 7x + 3 = 0
Math I
UNIT QUESTION: What is a
quadratic function?
Standard: MM2A3, MM2A4
Today’s Question:
How do you graph quadratic
functions in vertex form?
Standard: MM2A3.b.
3.2 Graphing Quadratic Functions
in Vertex or Intercept Form
• Definitions
• 3 Forms
• Steps for graphing each form
• Examples
• Changing between eqn. forms
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:
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
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!)
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
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.
Your Turn!
• Analyze and Graph:
y = (x + 4)2 - 3.
(-4,-3)
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
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?
(-1, 11)
(3,11)
X=1
(0,5)
(2,5)
(1,3)
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
x-coord. into the equation and solve for y.
If a is positive, parabola opens up
If a is negative, parabola opens down.
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 the vertex, use p 2 q
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.
(-2,0)
(4,0)
y  (1  2)(1  4)  (3)( 3)  9
• Vertex (1,9)
x=1
Now you try one!
y=2(x-3)(x+1)
• Open up or down?
• X-intercepts?
• Vertex?
• Axis of symmetry?
x=1
(-1,0)
(3,0)
(1,-8)
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
Challenge Problem
• Write the equation of the graph in vertex form.
y  3( x  2)2  4
Assignment
Day 1 -p. 65 #4,6,7,9,13,16
and Review for Quiz
Day 2 – p. 67 #4,5,7,9,11-14
We will not do intercept
form.