Reciprocal Functions

1
Chapter 1 Transformations
1.7A
Reciprocal Functions and Absolute Value Functions
1.7A.1
MATHPOWERTM 12, WESTERN EDITION
2
Reciprocal Functions
If f(x) x, then
represents a reciprocal function.
1.7A.2
3
Comparing y f(x) and
The graph of
can be obtained from the graph of
y f(x), using the following rules
When f(x)
then
y x
is less than -1
is between -1 and 0.
is -1.
is -1
is less than -1.
is between -1 and 0
may have a vertical asymptote.
is 0
is between 0 and 1
is greater than 1.
is 1
is 1.
y 1/x
is greater than 1
is between 0 and 1.
1.7A.3
4
Comparing y f(x) and
Where the value of the original function is
positive, the value of the reciprocal function
is positive.
Where the value of the original function is
negative, the value of the reciprocal function
is negative.
If the value of the original function increases
over an interval, the value of the reciprocal
function decreases over the same interval.
If the value of the original function decreases
over an interval, the value of the reciprocal
function increases over the same interval.
As the absolute value of the original function
increases, the absolute value of the reciprocal
function approaches 0.
1.7A.4
5
Sketching the Graphs of y f(x) and

• Draw a vertical asymptote at x 3.

• Plot the points that are the same for both.

• When f(x) lt -1, points are transformed to
• between -1 and 0 (visualize the idea of an
• inverse of the points).

• When f(x) is between -1 and 0, points are
• transformed to less than -1.

• When f(x) is between 0 and 1, points are
• transformed to greater than 1.

y f(x)

• When f(x) gt 1, points are transformed to
• between 0 and 1.

1.7A.5
6
from y f(x)
Sketching the Graph of
Sketch the graph of from the graph f(x) x2
3.
f(x) x2 3

• Since there are no real zeros for f(x),
• there are no vertical asymptotes for

• Since f(x) is always positive,

is always positive.
(0, 3)

• As x increases for f(x),

approaches 0.
1.7A.6
7
Sketching the Graph of y
from y f(x)

• There is a vertical asymptote at x 0.

• Plot (1, -1) and (-1, 1).

• The points on the graph of f(x)
• between the x-axis and y 1
• correspond to the points above
• the line y 1 on the reciprocal
• function. Points closer to the x-axis,
• correspond to points farther away
• from the line y 1.

y f(x)

• As the absolute value of
• the original function
• approaches 0, the absolute
• value of the reciprocal
• function increases.

• The points on the graph of f(x),
• that are greater than y 1,
• correspond to the points between
• the x-axis and y 1 on the
• reciprocal function. Points farther
• from the line of y 1, correspond
• to points closer to the x-axis.

1.7A.7
8
Finding the Equation and Graph of f(x) from
where f(x) is a quadratic function.
(0, 5)
(4, 5)
Given the graph of G(x), sketch the graph of
f(x) and find its equation.
f(x)
The y-intercept of G(x) is
The y-intercept of f(x) is 5.
The point (2, 1) is the same for both functions.
This point is the vertex of the parabola.
The graph of y x2 has been translated 2 units
to the right and 1 unit up
y a(x - 2)2 1
(2, 1)
Use the y-intercept of (0, 5)
G(x)
5 a(0 - 2)2 1 4
a(4) 1 a
Therefore, f(x) (x - 2)2 1.
1.7A.8
9
Assignment
Suggested Questions Pages 56 and 571, 2, 7,
9-11, 19, 21
1.7A.9
10
Chapter 1 Transformations
1.7B
Absolute Value Functions
1.7B.1
MATHPOWERTM 12, WESTERN EDITION
11
The Absolute Value Function
Recall that x is equal to x if x 0, and
equal to -x if x 0. When x 0, the graph
consists of a line defined by y x. When x 0,
the graph consists of a line defined by y -x.
f(x) x
1.7B.2
12
Sketching the Absolute Value of a Function
Sketch the graph of y 2x - 3 .
1. Sketch the graph of y 2x - 3.
y 2x - 3
2. The points on the graph of y 2x - 3
that are on or above the x-axis, are
invariant points.
3. The points on the graph of y 2x - 3
that are below the x-axis, are reflected in
the x-axis.
y 2x - 3
1.7B.3
13
Sketching the Absolute Value of a Function
Sketch the graph of y x2 - 5 .
1. Sketch the graph of y x2 - 5 .
2. Sketch the invariant points.
3. Reflect the points of y x2 - 5 that
are below the x-axis.
y x2 - 5
y x2 - 5
1.7B.4
14
Sketching the Absolute Value of a Function
y x2 - 4
1.7B.5
15
Assignment
Suggested Questions Pages 56 and 57 25, 29,
32-36, 39, 42-45, 50
1.7B.6