MATHPOWERTM 11, WESTERN EDITION

1
Chapter 3 Quadratic Functions
3.1
Graphing f(x) x2
q, f(x) ax2, and f(x) ax2 q
3.1.1
MATHPOWERTM 11, WESTERN EDITION
2
The Quadratic Function
A quadratic function is a function determined by
a second degree polynomial.
A quadratic function has a defining equation that
can be written in the form f(x) ax2 bx c
where a, b, and c are real numbers and a ? 0.
The graph of a quadratic function is a parabola
that has either a maximum (highest) point or a
minimum (lowest) point, called the vertex.
Every parabola has a vertical line, called the
axis of symmetry that passes through the vertex.
3.1.2
3
The Quadratic Function contd
The Parabola
Axis of symmetry
The graph opens upwards so there is a minimum
value.
Vertex
3.1.3
4
Graphing f(x) x2 q
Comparing f(x) x2 q with f(x) x2
(1,5)
(-1, 5)
(0, 4)
(1, 1)
(-1, 1)
(0,0)
x y -1 5 0 4 1 5
x y -1 1 0 0 1 1
(0, -4)
f(x) x2 - 4
f(x) x2
f(x) x2 4
(0, -4) x 0
(0, 4) x 0
Vertex is (0,0) Axis of Symmetry is x 0
Minimum value of y 0, when x 0
Minimum value of y 4, when x 0
Minimum value of y -4, when x 0
3.1.4
5
Graphing f(x) ax 2
Comparing f(x) ax2 with f(x) x 2
f(x) 0.5x2
f(x) 2x2
f(x) x2
x y 4 16 2 4 0 0 -2
4 -4 16
x y 4 32 2 8 0 0 -2
8 -4 32
x y 4 8 2 2 0 0 -2
2 -4 8
The y-coordinates of f(x) 2x2 are two times the
y-coordinates of f(x) x2. This results in a
vertical stretch factor of 2 and the parabola
narrows.
The y-coordinates of f(x) 0.5x2 are half the
y-coordinates of f(x) x2. This results in a
vertical compression factor of 0.5 and the
parabola widens.
3.1.5
6
Graphing f(x) ax 2 contd
f(x) 0.5 x2
f(x) x2
f(x) 2x2
x x2 2x2 0.5x2 0 0
0 0 1 1 2 0.5 2 4
8 2 3 9 18 4.5
x x2 2 x2 0 0 0 1
1 2 2 4 8 3
9 18
(1, 2)
Notice that the y-values are double.
Notice that the y-values are one-half.
(1, 1)
(1, 1)
The y-values of f(x) 2x2 are double that for
f(x) x2. Therefore, there is a vertical
stretch and the parabola narrows.
The y-values of f(x) 0.5x2 are half that for
f(x) x2. Therefore, there is a
vertical compression and the parabola widens.
(1, 0.5)
3.1.6
7
Graphing f(x) -x 2
f(x) x2
The graph of f(x) -x2 is the graph of f(x)
x2 reflected in the x-axis.
(-1, 1)
(1, 1)
(-1, -1)
(1,-1)
f(x) x2
3.1.7
8
Graphing a Quadratic Function
Graph f(x) x2 4.
The vertex is (0, 4).
f(x) x2 4
The axis of symmetry is x 0.
4
3
The parabola opens upward.
2
1
It has a minimum value of y 4, when x 0.
f(x) x2
The domain is all real numbers.
The y-intercept is (0, 4).
The range is y gt 4.
There are no x-intercepts.
3.1.8
9
Assignment
Suggested Questions
Pages 109 and 110 1, 4, 5, 14, 17, 19, 28-31, 43,
45, 48, 50, 70
3.1.9