Ch. 7 - Absolute Value and Reciprocal Functions Notes€¦ · Ch. 7.1 HW: p. 364 # 1 – 7 odd letters, 9, 12 . Pre-Calculus 11 Chapter 7 – Absolute Value and Reciprocal Functions

Pre-Calculus 11

Chapter 7 – Absolute Value and Reciprocal Functions

Created by Ms. Lee 1 of 26

Reference: McGraw-Hill Ryerson Pre-Calculus 11

First Name: ________________________ Last Name: ________________________ Block: ______

Ch. 7 – Absolute Value and Reciprocal Functions Notes

7.1 – ABSOLUTE VALUE 2

Ch. 7.1 HW: p. 364 # 1 – 7 odd letters, 9, 12 3

7.2 PRE-REQUISITES - GRAPH OF LINEAR FUNCTIONS 4

7.2 PRE-REQUISITES - GRAPH OF QUADRATIC FUNCTIONS 6

7.2 – ABSOLUTE VALUE FUNCTIONS 8

Ch. 7.2 HW: p. 375 #1 – 11 all 11

7.3 – ABSOLUTE VALUE EQUATIONS 12

Ch. 7.3 HW: p. 389 #1 – 6 odd letters, # 7, 11, 14 a, 17 15

7.4 – RECIPROCAL FUNCTIONS 16

Ch. 7.4 HW: p. 403 # 1 – 9, 11, 12, (need a graphing calculator),14, 15, 18 21

CH. 7 - REVIEW 22

Pre-Calculus 11




7.1 – Absolute Value

Definition:

Absolute value of x , denoted by ___________ can be thought of as:

� A distance or length between zero and x .

Eg: Distance between 0 and -5 = |-5| =

Distance between 0 and 10 = |10| =

� Therefore, output is always _____________

Examples:

1) Evaluate the following:

a) |-7|

b) |4| - |-16|

c) |5 – 3|

d) |2 – 10|

e) 3|-3|

f) -4|2 – 9|

g) |-2(5-7)2 + 3|

h) |-2+4(3)|

2) Use absolute value symbols to write an expression for the length of each horizontal or vertical line

segment. Determine each length.

a) A(12, 0) and B(-8, 9)

b) A(-1, -7) and B (-1, 15)

Pre-Calculus 11




3) On a particular day in Alberta, the temperature was °− 9 C in the morning. By afternoon, the

temperature was raised to °7 C and dropped to °− 5 C by night. Use absolute value symbols to

write an expression for the total change in temperature that day. What is the total change in the

temperature for the day?

4) Evaluate.

a) 2|3-5| + 3|4-1|

b) (-3|-4| - 2|-4|)

c) |105|

|64|

−−

+−

c) |53||24|

|104||3|

+−−−

−+−+5

Ch. 7.1 HW: p. 364 # 1 – 7 odd letters, 9, 12

Pre-Calculus 11




7.2 Pre-requisites - Graph of Linear Functions

Graph each linear function:

1) y = 2

1x + 5

2) y = 43

2−x

3) y = 14 +− x

4) y = 23 −− x

5) y = 65

4−x

6) y = 23

1−− x

Pre-Calculus 11




1) y = 5

2) y = x4

3

3) 2−=x

4) zs 13 += xy

5) 32 −−= xy

6) 2−=y

Pre-Calculus 11




7.2 Pre-requisites - Graph of Quadratic Functions

Graph each quadratic function:

1) 4)1(3 2 −−= xy

2) 52 2 +−= xy

3) 1)3(2

1 2−−= xy

4) 5)2(3

1 2++−= xy

5) 163 2 −+= xxy

6) 342 ++−= xxy

Pre-Calculus 11




Graph each quadratic function in factored form:

1) )3)(1( +−= xxy

2) )2)(2( +−−= xxy

3) 2)2( −= xy

4) 2)3( += xy

5) )4)(2(2 ++= xxy

6) )5)(1( +−−= xxy

Pre-Calculus 11




7.2 – Absolute Value Functions

Definition:

An absolute value function is a function that involves the absolute value of a variable.

A piecewise function is a function composed of two or more separate functions or pieces, each with its

own specific domain, that combine to define the overall function.

Example: Absolute Value Function of the form, || baxy +=

1) Graph xxf =)( (use dotted line to represent this function)

Graph |)(|)( xfxg = on the same grid.

Define ||)( xxg = as a piecewise function.

2) Graph 3)( −= xxf (use dotted line to represent this function)

Graph |)(|)( xfxg = on the same grid.

Define |3|)( −= xxg as a piecewise function.

Pre-Calculus 11




3) Graph the absolute value function, |3| += xy .

Define |3| += xy as a piecewise function.

4) Graph the absolute value function, |42|)( −= xxf .

Define |42|)( −= xxf as a piecewise function.

5) Given )(xfy = below, graph |)(| xfy = .

a)

b)

Pre-Calculus 11




Example: Absolute Value Function of the form, || 2 cbxaxy ++=

6) Graph the absolute value function, |82|)( 2 ++−= xxxf .

1) Write 822 ++−= xxy in factored form.

2) Determine the zeros of the function

3) Find the vertex.

4) Sketch 822 ++−= xxy (in dotted line)

5) Take the absolute value of 822 ++−= xxy and

sketch )(xf

7) Express the absolute value function, |82|)( 2 ++−= xxxf as a piecewise function.

Pre-Calculus 11




8) Graph the absolute value function, |44

1|)( 2 −= xxf . Express the absolute value function,

|44

1|)( 2 −= xxf as a piecewise function.

9) Graph the absolute value function, |4)1(2

1|)( 2 −+−= xxf . Express the absolute value function,

|4)1(2

1|)( 2 −+−= xxf as a piecewise function.

Ch. 7.2 HW: p. 375 #1 – 11 all

Pre-Calculus 11




7.3 – Absolute Value Equations

1)

Solve 5|2| =−x by graphing

Solve 5|2| =−x algebraically.

Case 1:

Case 2:

2)

Solve 53|42| +=+ xx by graphing

Solve 53|42| +=+ xx algebraically.

Case 1:

Case 2:

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3)

Solve xx −=− 4|52| by graphing

Solve xx −=− 4|52| algebraically.

Case 1:

Case 2:

4)

Solve 1|2| 2 =− xx by graphing

Solve 1|2| 2 =− xx algebraically.

Case 1:

Case 2:

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5) Solve 2|2| 2 =−− xx

6) Solve 158|5| 2 +−=− xxx

Pre-Calculus 11




7) Solve 13|32| −=− xx

8) Solve 4|)3)(1(| =+− xx

9) Determine an absolute value equation in the form |ax + b| = c given its solutions on the number

line.

Ch. 7.3 HW: p. 389 #1 – 6 odd letters, # 7, 11, 14 a, 17

Pre-Calculus 11




7.4 – Reciprocal Functions

Definition:

Asymptote: A line that a curve approaches but never crosses (touches) as one of the variables

approaches some particular values.

Ex:

We can have both horizontal and vertical asymptotes

We can just have horizontal

asymptotes

Reciprocal:

What is the reciprocal of 4? What is the reciprocal of 5

1?

Given the function, )(xfy = . What is the reciprocal of this function?

Reciprocal Function: A function defined by )(

1

xf where ≠)(xf 0.

Functions:

)(xfy = Reciprocal Functions:

)(

1

xfy =

Domain of the reciprocal

xy =

52 += xy

Pre-Calculus 11




42 −= xy

62 −+= xxy

42 += xy

How to get horizontal asymptotes:

Examine the reciprocal functions, as x approaches ∞± , the y-value approaches ________.

For any reciprocal functions, )(

1

xfy = , would the y-value ever be zero? ______

Therefore, the range of a reciprocal function is: __________________________________.

This means, there will be a ___________________ asymptote defined by ______________.

How to obtain vertical asymptotes:

Where there is a non-permissible value, there will be a ______________________ asymptote.

Examples:

1) Graph 13)( −== xxfy (in red) and its reciprocal (in blue) on the same grid.

x )(xf )(

1

xf

Equation of V.A: Equation of H.A:

Pre-Calculus 11




2) Graph xxfy 3)( −== (in red) and its reciprocal (in blue) on the same grid.

x )(xf )(

1

xf


3) Graph 5)( 2 −== xxfy (in red) and its reciprocal (in blue) on the same grid.

x )(xf )(

1

xf


Pre-Calculus 11




4) Graph: )(xfy = = 32 +x (in red) and its reciprocal (in blue) on the same grid

x )(xf )(

1

xf


5) State the equation(s) of the vertical asymptote(s) for each function.

Reciprocal Functions Non-permissible Values Equations of vertical asymptotes

42

1)(

+=

xxf

)3)(1(

1)(

−+=

xxxf

2)12(

1)(

+

=

xxg

9

1

2−

=

xy

3

1)(

2+

=

xth

Pre-Calculus 11




6) Given the graph of )(xfy = , and sketch the graph of the reciprocal function )(

1

xfy = .

7) The calculator screen gives a function table for 4

1)(

2−

=

xxf . Explain why there is an ERROR

statement for 2±=x .

8) State x-intercept(s) and the y-intercept of each function.

a) 3

1)(

+=

xxf

Pre-Calculus 11




b) 7

1

2−

=

xy

c) 103

1

2−+

=

xxy

9) Given the graph of )(xfy = , and sketch the graph of the reciprocal function )(

1

xfy = .

Ch. 7.4 HW: p. 403 # 1 – 9, 11, 12, (need a graphing calculator),14, 15, 18

Pre-Calculus 11




Ch. 7 - Review

Multiple Choice Questions:

1. The zero(es) of xy 21−= occurs when

a) 2

1=x b)

2

1−=x

c) 2=x d) 2−=x

2. The zero(es) of 422 −−= xxy occurs when

a) 101±=x b) 51±=x c) 1±=x d) 1=x

3) Which of the following is the graph of )3)(1()( −+= xxxf .

a)

b)

c)

d)

4) Given a linear function, xxf −= 2)( , which of the following is the graph of the reciprocal

function, )(

1

xfy = ?

a)

b)

c)

d)

5) Determine the equations of the vertical asymptotes of 9

1

2−

=

xy .

a) 3±=x b) 3±=x c) 3=x only

d) No V.A.

Pre-Calculus 11




Written Response: Show all your work clearly for full marks.

6) Evaluate |3|4|2| −+− xx if 1−=x .

7) Write |21|)( xxf −= as a piece-wise function. The graph of )(xfy = is shown below.

[2 marks]

8) Write |)3)(1(|)( +−= xxxf as a piece-wise function.

[2 marks]

9) Write |13|)( 2 −+= xxxf as a piece-wise function. The graph of )(xfy = is shown below.

[2 marks]

10) Graph each absolute value function, [2 marks]

a) |53| −= xy

b) |3)2(| 2 +−−= xy

Pre-Calculus 11




Solve 7|32| =−x . [3 marks]

Domain for Case 1: _________________________

Domain for Case 2: _________________________

Therefore, solution(s) is/are: ___________________________

11) Solve 1|)1)(1(| +=−+ xxx . [3 marks]

Domain for Case 1: _________________________

Domain for Case 2: _________________________

Therefore, solution(s) is/are: ___________________________

Pre-Calculus 11




12) Solve by graphing 3|32| 2 +=++ xxx [3 marks]

Solution(s): ____________________

13) Given the graph of )(xfy = , sketch the graph of )(

1

xfy = . Clearly show invariant points and a

few other points. Also draw the asymptote(s) in dotted line(s). [2 marks]

Pre-Calculus 11




14) Solve 622 =− xx [3 marks]

Domain for Case 1: _____________________

Domain for Case 2: _____________________

Therefore, solution(s) is/are: __________________________

Note: Write the solutions in their simplest expressions. Do not round.

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Ch. 7 - Absolute Value and Reciprocal Functions Notes€¦ · Ch. 7.1 HW: p. 364 # 1 – 7 odd letters, 9, 12 . Pre-Calculus 11 Chapter 7 – Absolute Value and Reciprocal Functions