Chapter 7: Absolute Value and Reciprocal Functions
A1 – Demonstrate an understanding of the absolute value of real numbers.
C2 – Graph and analyze absolute value functions (limited to linear and quadratic
functions) to solve problems.
C11 – Graph and analyze reciprocal functions (limited to the reciprocal of linear and
quadratic functions).
Section 7.1 Absolute Value
Section 7.4 Reciprocal Functions (Part
1)
-Page 363
#1-7
-P403
# 1, 2ab, 3ab, 5ab, 6a, 7bcd, 10a
-Worksheet
-Mid chapter checkpoint
Section 7.2 Absolute Value Functions
Section 7.4 Reciprocal Functions (Part
(Part 1)
2)
-Page 375
#1a, 2-5, 6ace, 9,11ab, 12
-P403
# 2cd, 3cd, 6bc, 8bd, 10b
Section 7.2 Absolute Value Functions
(Part 2)
Word Problems
-Page 375
#7, 8aef, 10, 11cd, 13
-P403
# 5, 9, 18
-checkpoint on 7.1 and 7.2 (P1)
Section 7.3 (Part 1) Solving Absolute
Value Equations Algebraically
Review
-Page 389
#4,5,6b
-checkpoint on 7.2 (P2)
Section 7.3 (Part 2) Solving Absolute
Value Equations by graphing
-worksheet
-checkpoint on 7.3 (P1)
pg. 1
Test
Example 2: Write the following in order from least to greatest.
6.5 , 5 , 4.75 ,  3.4 , 
12
, 0.1
5
Example 3: Evaluate each expression. Use BEDMAS. Absolute value is treated as brackets.
a)
4  7
b)
8  9  21
c)
6  2 6  11
d)
2 3  6  2 22  1
Practice: p. 363 # 1-7 and look at lines worksheet
PS. Need graph paper for the next lessons
7.2 Absolute Value Functions – Part 1
An absolute value function is a function that involves the absolute value of a ____________.
Recall: The absolute value of 'x' is defined as
y x
and can be written as
 x, if x  0
y
 x, if x  0
Since the function is defined by two different rules for each interval in the domain, this is an example
of a ____________________.
Example 1: Sketch the graph of
y x
Example 4: The point (-6, -4) is on the graph of y = f(x). Identify the corresponding point on the graph
of
y  f (x) .
Example 5: Graph Consider the function
y  3x  2 .
(a) Method 1:
(b) Method 2:
(c) State the x-intercept and y-intercept
(d) State the domain and range.
(e) Express as a piecewise function.
Example 4: Consider the function
y   2x  3 .
(a) Determine the x-intercept and the y-intercept.
(b) Sketch the graph.
Method 1: Table of Values
Method 2: Transformations
(c) State the domain and range.
(d) Express as a piecewise function.
Example 5: Write the piecewise function that represents the following graph
Practice: p. 375 #1a, 2-5, 6ace, 9, 11ab,12
y  3x  6
7.2 Absolute Value Functions: Part II
Example 1: Given the graph of y = f(x). On the same set of axes, sketch the graph of y 
f (x) .
Example 2: What piecewise function could you use to represent each graph of an absolute value
function?
(a)
y  3( x  4) 2  3
(b)
y  2 x2  8
Example 3: Consider the function
y   x2  2x  8
.
(a) Determine the x-intercept and the y-intercept.
(b) Sketch the graph.
(c) State the domain and range.
(d) Express as a piecewise function.
Practice: p. 375 #7, 8aef, 10, 11cd, 13
7.3 Part I - Solving Absolute Value Equations Algebraically
Example 1: Solve and check:
2x  6  4
Case 1: the expression inside the absolute value
is POSITIVE
Example 2: Solve and check:
Case 1:
Case 2: the expression inside the absolute value
is NEGATIVE
2 x  5  5  3x
Case 2:
Example 3: Solve and check:
3x  4  12  9
Example 4: Solve and check:
 x2  4  3
Case 1:
Practice: p. 389 # 4, 5, 6b
Case 2:
7.3 Part 2 - Solving Absolute Value Equations by Graphing
An Absolute Value Function is written in the form _________________.
Absolute Value means ______________________________________________________________.
Basic Absolute value function: Graph:
y x
y
6
5
4
3
2
1
-6
-5
-4
-3
-2
-1
x
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
Example 1: Review – graph each of the following absolute value functions.
a)
y  x2
b)
y  2x  3
y
y
6
6
5
5
4
4
3
3
2
2
1
-6
c)
-5
-4
-3
-2
1
x
-1
1
2
3
4
5
6
-6
-5
-4
-3
-2
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
y  x2  4
d)
x
-1
6
5
5
4
4
3
3
-3
-2
-1
5
6
2
1
-4
4
y
6
2
-5
3
y  x2  1
y
-6
2
1
x
1
2
3
4
5
6
-6
-5
-4
-3
-2
-1
x
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
2
3
4
5
6
Example 2: Solve the following equations by graphing:
a) |𝑥 − 3| = 4
Steps:
1. Graph the left function y1
2. Graph the right function y2
3. Find where the graphs intersect
y
6
5
4
3
2
1
-6
-5
-4
-3
-2
x
-1
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
b) |𝑥 − 4| = 2𝑥 + 1
c) |𝑥 + 3| = |𝑥 − 4|
y
y
6
6
5
5
4
4
3
3
2
2
1
1
-6
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
x
x
2
3
4
5
6
7.3 part 2 - Solving Absolute Value Equations by Graphing Worksheet
1. Solve each of the following by graphing.
a) |2𝑥 − 5| = 5 − 3𝑥
b) |𝑥 − 3| = |𝑥 + 1|
y
y
-6
-5
-4
6
6
5
5
4
4
3
3
2
2
1
1
-3 -6 -2 -5 -1 -4
-1
-3 1 -2 2 -1 3
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
x
x
4 1 5 2 6 3
4
5
6
c) |𝑥| + 8 = 12
d) |𝑥| = −5
y
y
6
6
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
1
x
-1
1
2
3
4
5
6
-6
-4
-6
-3
-3
-2
-1
-1
-2
-3
-3
-4
-4
-5
-5
-6
-6
f) |𝑥 2 − 2𝑥 + 2| = 3𝑥 − 4
y
-5
-2
6
6
5
5
4
4
3
3
2
2
1
1
-4
-1
-3
-2
1
-1
2
3
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
x
x
41
52
63
4
5
6
x
1
-2
y
-5
-4
-1
e) |𝑥 + 3| = 2
-6
-5
2
3
4
5
6
2. Solve each of the following by graphing.
a) |𝑥 2 − 4| = 𝑥 − 2
b) |𝑥| = 𝑥 2
y
y
6
6
5
5
4
4
3
3
2
2
1
-6
-5
-4
-3
-2
1
x
-1
1
2
3
4
5
-6
6
-5
-4
-3
-2
-1
1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
c) |2𝑥 2 − 2| = −𝑥 − 1
x
-4
-3
-2
6
5
5
4
4
3
3
2
2
-1
1
2
3
4
5
6
-6
-4
-3
-6-2
-5-1
-5
-4
-3
-2
-1
6
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
f) |𝑥| = |−3|
y
6
6
5
5
4
4
3
3
2
2
1
1
-4
-1
-3 1
-2 2
-1 3
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
x
x
4
15
26
3
4
5
6
x
1
-1
y
-5
5
1
x
e) |𝑥 + 1| = |𝑥 − 1|
-6
4
y
6
1
-5
3
d) |𝑥 2 − 1| = −1
y
-6
2
2
3
4
5
6
7.4 Reciprocal Functions – Part 1
Recall: For any real number a its reciprocal is __________
e.g.
For a function
y  f (x) its reciprocal is defined by y  1
f ( x)
Example 1: Given the function
a) y  2 x  5
c)
y  x 2  3x  8
provided that ______________.
y  f (x) determine the corresponding reciprocal function.
b) y  7 x  9
d)
y   x2  4
Example 2: What are the non-permissible values for the following functions?
a) f ( x ) 
1
2x  8
b) f ( x) 
1
x  2 x  48
2
Example 3: Sketch the graphs of y  x and it’s reciprocal.
x
o
What is the equation of the reciprocal? ____________________
o
What is the non-permissible value?
yx
y
1
x
____________________
9
8
7
6
5
4
3
2
1
9 8 7 6 5 4 3 2 11
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
An ________________________ is a line that a graph approaches more and more closely the further the
graph is followed. You can have ________________________ and ________________________ asymptotes.
Properties of Reciprocal Functions:
Original Function f ( x )
Reciprocal Function
Example 4: Given the following graph sketch the reciprocal.
7
6
5
4
3
2
1
7
6
5
4
3
2
1
1
1
2
3
4
5
6
7
2
3
4
5
6
7
8
1
f ( x)
Example 5: Given the function
y  2x  8
a) Determine the reciprocal equation.
b) Determine the equation of the vertical asymptote of the reciprocal function.
c) Graph the function and its reciprocal.
9
8
7
6
5
4
3
2
1
9 8 7 6 5 4 3 2 11
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
Example 6: The graph of the reciprocal function of the form y 
1
where a and b are non-zero
ax  b
constants, is shown.
a) Sketch the graph of the
original function.
7
6
5
4
3
2
(3, ½)
1
7
6
5
4
3
2
1
1
1
2
3
4
5
6
7
b) Determine the original function y  f ( x) .
Practice: p. 403 #1, 2ab, 3ab, 5ab, 6a, 7bcd, 10a
2
3
4
5
6
7
8
7.4 Reciprocal Functions – Part 2
Recall: A reciprocal function has a vertical asymptote whenever the _______________________ is equal
to zero.
Example1: Find the equation of the vertical asymptotes for the following.
a) f ( x) 
1
x  x  30
b) y 
2
1
x  3x
2
Example 2: Given the following graph sketch the reciprocal.
7
6
5
4
3
2
1
7
6
5
4
3
2
1
1
1
2
3
4
5
6
7
2
3
4
5
6
7
8
Example 3: Given the function f ( x)  x  6 x  8 .
2
a) Determine the reciprocal function.
b) Determine the equation(s) of the vertical asymptote of the reciprocal function.
c) Sketch the graph of the function and its reciprocal.
7
6
5
4
3
2
1
7
6
5
4
3
2
1
1
1
2
3
4
5
6
7
2
3
4
5
6
7
8
Example 4: The graph of the reciprocal function of the form y 
1
is shown.
f ( x)
a) Sketch the graph of the original function.
9
8
7
6
5
4
3
2
1
9
8
7
6
5
4
3
2
1
1
1
2
3
4
5
6
7
8
9
b) What is the equation of the original function?
Practice: p.403 #2cd, 3cd, 6bc, 8bd, 10
2
3
4
5
6
7
8
9
10