9.1 Graph Quadratic Functions
Alg. I
• Definitions
• 3 forms for a quad. function
• 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 (minimum) or
• highest point (maximum)
of a parabola.
Vertex
Axis of symmetry• The vertical line through the vertex of the
parabola.
Axis of
Symmetry
Standard Form Equation
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y=ax2 + bx + c
If a is positive, u opens up
If a is negative, u opens down
b
The x-coordinate of the vertex is at 2a
To find the y-coordinate of the vertex, plug the xcoordinate into the given eqn.
The axis of symmetry is the vertical line x= 2ab
Choose 2 x-values on either side of the vertex xcoordinate. Use the eqn to find the corresponding yvalues.
Graph and label the 5 points and axis of symmetry on a
coordinate plane. Connect the points with a smooth
curve.
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Example 1: Graph y=2x2-8x+6
• a=2 Since a is positive
the parabola will open up.
• Vertex: use x  2ab
b=-8 and a=2
 (8) 8
x
 2
2(2)
4
y  2(2) 2  8(2)  6
y  8  16  6  2
Vertex is: (2,-2)
• Axis of symmetry is the
vertical line x=2
•Table of values for other
points:
x y
0 6
1 0
2 -2
3 0
4 6
* Graph!
x=2
Now you try one!
y=-x2+x+12
* Open up or down?
* Vertex?
* Axis of symmetry?
* Table of values with 5 points?
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!)
Example 2: Graph
y=-.5(x+3)2+4
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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 5 points?
Intercept Form Equation
y=a(x-p)(x-q)
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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,
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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?
Changing from vertex or intercepts
form to standard form
• The key is to FOIL! (first, outside, inside,
last)
• Ex 4: y=-(x+4)(x-9)
Ex 5: y=3(x-1)2+8
Assignment
(-1, 11)
(3,11)
X=1
(0,5)
(2,5)
(1,3)
(.5,12)
(-1,10)
(2,10)
(-2,6)
(3,6)
X = .5
x=1
(-1,0)
(3,0)
(1,-8)
=-(x2-9x+4x-36)
=-(x2-5x-36)
x+1)+8
y=-x2+5x+36
=3(x-1)(x-1)+8
=3(x2-x=3(x2-2x+1)+8
=3x2-6x+3+8
y=3x2-6x+11