Chapter 7

ABSOLUTE VALUE AND RECIPROCAL

FUNCTIONS

EXAMPLE

How many triangles are in the diagram below?

Chapter 77.1 – ABSOLUTE

VALUE

ABSOLUTE VALUE

For a real number a, the absolute value is written as |a| and is a positive number. For example:

|5| = 5|–5| = 5

Absolute value can be used to represent the distance of a number from zero on a real-number line.

Evaluate:

|3| |–7|

EXAMPLE

Evaluate the following:a) |4| – |–6| b) 5 – 3|2 – 7| c) |–2(5 – 7)2 + 6|

a) |4| – |–6| = 4 – 6 = –2

b) 5 – 3|2 – 7| = 5 – 3|–5| = 5 – 3(5) = 5 – 15 = –10

c) Try it!

Independent Practice

PG. 363-367 #1, 6, 7(A,C,E), 11, 16

HANDOUT

Answer the questions on the “Investigating Absolute Value Functions” worksheet to the best of your ability.

Chapter 77.2 – ABSOLUTE

VALUE FUNCTIONS

ABSOLUTE VALUE FUNCTIONS

For what values of x is the function y = |x| equivalent to y = x?

when x ≥ 0When x < 0, what is the function represented by y = |x|?

y = –x

We can write this as a piecewise function:

EXAMPLE

Consider the absolute value function y = |2x – 3|.a) Determine the y-intercept and the x-intercept.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function.

a) The y-intercept is at x = 0. y = |2x – 3|y = |2(0) – 3|y = |–3|y = 3The y-intercept is (0, 3).

The x-intercept is at y = 0.0 = |2x – 3|0 = 2x – 3x = 3/2The x-intercept is at (3/2, 0).

b) x y

-1 5

0 3

3/2 0

3 3

4 5

EXAMPLE

Consider the absolute value function y = |2x – 3|.a) Determine the y-intercept and the x-intercept.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function.

x y

-1 5

0 3

3/2 0

3 3

4 5

b) c) D: {x | x E R} R: {y | y ≥ 0, y E R}

d) The equation on the right is just y = 2x – 3.

What’s the one on the left?

It’s just –(2x – 3)!

The x-intercept is call an invariant point because it’s a part of both functions.

EXAMPLE

Consider the absolute value function f(x) = |–x2 + 2x + 8|.a) Determine the y-intercept and the x-intercepts.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function

a) The y-intercept is at x = 0.

f(0) = |–(0)2 + 2(0) + 8| = |8| = 8

The x-intercepts are when y = 0. 0 = |–x2 + 2x + 8| 0 = –x2 + 2x + 8 0 = –(x – 4)(x + 2) x = 4 x = –2

b) What’s the vertex of the function

f(x) = –x2 + 2x + 8

Use your calculator, or complete the square:

f(x) = –(x2 – 2x) + 8 f(x) = –(x2 – 2x + 1 – 1 ) + 8 f(x) = –(x2 – 2x + 1) + 1 + 8 f(x) = –(x – 1)2 + 9 The vertex is (1, 9)

EXAMPLE

Consider the absolute value function f(x) = |–x2 + 2x + 8|.a) Determine the y-intercept and the x-intercepts.b) Sketch the graph.c) State the domain and range.d) Express as a piecewise function

Recall: y-intercept is (0, 8) x-intercepts are (4, 0) and

(–2, 0) Vertex is (1, 9)c) D: {x | x E R} R: {y | y ≥ 0, y E R}

d)

Independent Practice

PG. 375-379, #2, 5, 7, 10, 12-14.

Chapter 77.3 - ABSOLUTE

VALUE EQUATIONS

ABSOLUTE VALUE EQUATIONS

When solving equations that involve absolute value equations you need to consider two cases:

Case 1: The expression inside the absolute value symbol is positive or zero.

Case 2: The expression inside the absolute value symbol is negative.

EXAMPLE

Solve: |x – 3| = 7

Consider the equation as a piecewise function:

Case 1:

x – 3 = 7 x = 10

Case 2:

–(x – 3) = 7 x – 3 = –7 x = –4

The solution is x = 10 or x = –4.

TRY IT

Solve |6 – x| = 2

EXAMPLE

Solve |2x – 5| = 5 – 3x

Consider:

What is the x-intercept of y = 2x – 5? 0 = 2x – 5 5 = 2x x = 5/2

Case 1: (x ≥ 5/2)

2x – 5 = 5 – 3x 5x = 10 x = 2

Case 2: (x < 5/2)

–(2x – 5) = 5 – 3x –2x + 5 = 5 – 3x x = 0

EXAMPLE

Solve: |3x – 4| + 12 = 9

|3x – 4| = –3

Is there any possible way that the absolute value of something is equal to –3?

No solution.

EXAMPLE

Solve: |x – 10| = x2 – 10x

Independent Practice

PG. 389-391, #4, 5, 6, 9, 11, 22, 23

Chapter 77.4 – RECIPROCAL

FUNCTIONS

EXAMPLE

Sketch the graphs of y = f(x) and its reciprocal function y = 1/f(x), where f(x) = x. Examine how the functions are related.

x y = x y = 1/x

–5 –5 –1/5

–2 –2 –1/2

–1 –1 –1

–1/2 –1/2 –2

–1/10 –1/10 –10

0 0 Undef.

1/10 1/10 10

1/2 1/2 2

1 1 1

2 2 1/2

RECIPROCAL FUNCTIONS

An asymptote is a line whose distance from a curve approaches zero.

This graph has two pieces, that both approach the vertical asymptote, which is defined by the non-permissible value of domain of the function, and a horizontal asymptote, defined by the value that is not in the range of the function.

What is the vertical asymptote?

What is the horizontal asymptote?

EXAMPLE

Consider f(x) = 2x + 5.a) Determine its reciprocal function y = 1/f(x).b) Determine the equation of the vertical asymptote of the reciprocal

function.c) Graph the function y = f(x) and its reciprocal function y = 1/f(x).

a) The reciprocal function is:

b) The vertical asymptote is always the non-permissible values of the function.

2x + 5 = 0 2x = –5 x = –5/2

There is a vertical asymptote at x = –5/2

c)Characteristic

f(x) = 2x +5

f(x)=1/(2x + 5)

x-intercept/asymptotes

x-intercept at x = –5/2

Asymptote at x = –5/2

Invariant points

2x + 5 = 1 x = –2 (–2, 1)

(–2,1)

2x + 5 = –1 x = –3 (–3,–1)

(–3,–1)Invariant points are at y = 1 and y = –1.

EXAMPLE

Characteristic

f(x) = 2x +5

f(x)=1/(2x + 5)

x-intercept/asymptotes

x-intercept at x = –5/2

Asymptote at x = –5/2

Invariant points

2x + 5 = 1 x = –2 (–2, 1)

(–2,1)

2x + 5 = –1 x = –3 (–3,–1)

(–3,–1)

Consider f(x) = 2x + 5.a) Determine its reciprocal function y = 1/f(x).b) Determine the equation of the vertical asymptote of the reciprocal

function.c) Graph the function y = f(x) and its reciprocal function y = 1/f(x).

EXAMPLE

Consider f(x) = x2 – 4.a) What is the reciprocal function of f(x)?b) State the non-permissible values of x and the equation(s) of the vertical

asymptote(s) of the reciprocal function.c) What are the x-intercepts and y-intercepts of the reciprocal function?d) Graph the functions.

a)

b) What are the non-permissible values? x2 – 4 = 0 (x – 2)(x + 2) = 0 x = 2 x = –2

The vertical asymptotes are at x = ±2

c) How can I find the x-intercept of the the reciprocal? Let f(x) = 0

There is no solution, so there is no x-intercept.

y-intercept:y-intercept is y = –1/4

Independent practice

PG. 403-408, #3, 5, 7, 8, 9, 10, 12