Ch 7 Notes - WordPress.com · Ch. 7: Absolute Value and Reciprocal Functions Loo/Lee/Ko Page 26 of 41 Example 2: a) Graph the reciprocal function b) Determine the equation of any

Loo/Lee/Ko

Pre-Calculus11

ABSOLUTEVALUE&

RECIPROCALFUNCTIONS Chapter 7

7.1 AbsoluteValue............................p.1

7.2 GraphingPre-requisites...................p.5

7.2 AbsoluteValueFunctions.................p.10

7.3 AbsoluteValueEquations:Part1..........p.15

7.3 AbsoluteValueEquations:Part2..........p.19

7.4 ReciprocalFunctions:Part1...............p.24

7.4 ReciprocalFunctions:Part2...............p.29

Ch.7Review....................................p.33

Pre-Calculus 11 Date: ____________________

Ch. 7: Absolute Value and Reciprocal Functions Loo/Lee/Ko Page 1 of 41

7.1 Absolute Value

Absolute value of a number represents the distance that number is from zero on the number line,

regardless of the direction.

Example 1: 5 5 because both are 5 units away from zero.

Notation: Vertical bars are used around a number or expression to represent its absolute value,

ie: ,

Solutions for Absolute Values The Absolute value is always zero or positive.

- The absolute value of a positive number is positive:

- The absolute value of a negative number is positive:

- The absolute value of zero is zero:

Example 2: Evaluate.

a) b) c) d)

e) 3 3 f) 4 2



g) h) 3| 4| 2| 4|

Example 3: Use absolute value symbol(s) to write an expression for the length of the vertical line

segment and determine the length for A and B

Example 4: Order the numbers from greatest to least.



Example 5: The Yukon Quest dog sled race runs between Fairbanks, Alaska, and Whitehorse, Yukon Territory, a distance of more than 1000 mi. It lasts for 2 weeks. The elevation at Fairbanks is 440 ft, and the elevation at Whitehorse is 2089 ft. a) Determine the net change in elevation from Fairbanks to Whitehorse. b) The race passes through Central, at an elevation of 935 ft; Circle City, whose elevation is 597 ft; and Dawson City, at an elevation of 1050 ft. What is the total change in elevation from Fairbanks to Whitehorse, passing through these cities? Assignment Page 363, #1abc, 2, 4, 5ac, 6, 7ace, 8, 11, 15





7.2 Pre-requisites - Graph of Linear Functions Graph each linear function:

1) y = 21

x + 5

2) y = 432

x

3) y =

4) y =

5) y = 654

x

6) y =



7) y = 5

8) y = x43

9)

10.

11.

12)



7.2 Pre-requisites - Graph of Quadratic Functions Graph each quadratic function:

1)

2) 52 2xy

3) 4)1(3 2xy

4) 1)3(21 2xy

5) 163 2 xxy



Graph each quadratic function in factored form:

6) 342 xxy

7)

8)



7.2 Absolute Value Functions

Absolute Value Function

An absolute value function has the form , where is a function.

Definitions

Critical point The graph of changes directions at this point. This point is at the x-

intercept of the function .

Invariant point a point that remains unchanged when a transformation is applied to it.

Piecewise function a function composed of two or more separate functions or pieces, each with its

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

Definition: The absolute value of , , can be written as a piecewise function:

Graphically: Comparing and

Note: the critical point or vertex

of this absolute value is .



In order to graph the absolute value of a function , you must first graph the original function

.

Points on the graph whose y-coordinates are positive or equal to zero, (above and including the

x-axis) will remain the same. Connect the points with a solid line.

Points on the original function whose y-coordinates are negative (below the x-axis) will be

reflected across the x-axis and made positive. Connect the points with a dotted line.

Example 1: Consider the absolute value function .

a) Determine the x and y intercepts of the function

b) Sketch the graph

c) State the domain and range.

d) Express as a piecewise function.



Example 2: For the absolute value function,

a) Determine the -intercept and the -intercept.

b) Sketch the graph.

Method 1 Method 2

c) State the domain and range. d) Express as a piecewise function.



Example 3: a) Graph the absolute value function,

b) Express the function in piecewise notation.

Example 4: a) Graph the absolute value function,

b) Express the function in piecewise notation.

Assignment Page 375, #5b, 6ace, 7b, 8acf, 9b, 10c, 11, 13, 14, 17





7.3 Absolute Value Equations: Part 1 Absolute value equation

An equation that includes the absolute value of an expression involving a variable.

To solve an absolute value equation, we have two methods which we can apply.

Graphically: graph both sides of the equation as two separate functions and look for the

x- coordinates of the points of intersection.

Algebraically:

1. Use definition of absolute value to find the critical point(s) of the function.

2. Find out

3. Break up the domain into possible regions.

4. Go through and do the calculations for all the possible cases.

5. Solve and check to see if your answer is valid. Eliminate extraneous roots

Example 1:

a) Solve the equation by graphing.

b) Solve the equation algebraically.



Example 2:

a) Solve the equation by graphing.


Example 3: How many possible solutions can exist between a linear and absolute value function?

Demonstrate all possibilities using diagrams.



Example 4: Solve 5 4 1x x algebraically.

Example 5: Solve

Assignment Page 389, #2cd, 4acd, 5bde, 13, 14





7.3 Absolute Value Equations: Part 2

We can solve absolute value equations involving quadratics in the same way we do for absolute value

equations involving linear expressions: graphically or algebraically.

Example 1: a) Graph each side of the equation

, on the grid.

b) Write the equation of the absolute value

function in piecewise notation.

c) Why is it difficult to determine the solution of this equation graphically?

d) Solve the equation algebracially



Example 2: a) Graph


Example 3: Determine the number of possible solutions between the absolute value of a quadratic

function and a linear function. Sketch diagrams of your results.



Example 4: Solve algebraically.

Example 5: Solve 25 8 15x x x

Assignment Page 389, #6, 22, 23, Worksheet: #24



7.3 Absolute Value Equations Worksheet

24. Solve algebraically. a) b) c) d)

Answers: 24a) b)

c) d)





7.4 Reciprocal Functions: Part 1

What is a reciprocal? ______________________________________________________________

Review Exercise: Find the reciprocals for the following.

a) b) c) d)

Reciprocal Function

Given a function , the reciprocal function has the form , where is a function,

and . **Need to know:**

1. What is the reciprocal of 0?

2. What are the only two numbers that are equal to their own reciprocals? Definitions: Asymptote a line whose distance from a given curve approaches zero. Vertical asymptote for reciprocal functions, occur at the non-permissible values of the function. The line is a vertical asymptote if the curve approaches the line more and more closely as approaches , and the values of the function increase or decrease without bound as approaches .

Horizontal asymptote describes the behavior of a graph when is very large. The line

is a horizontal asymptote if the values of the function approach when is very large.

Invariant point a point that remains unchanged when a transformation is applied to it.

Just as the reciprocal of a number is , provided that , similarly,

the reciprocal of a function is the function provided that 0f x .



Ex. 1: The most basic reciprocal function is x

y1

which is the reciprocal of the identity line, .

Characteristic

Domain

Range

End Behaviour

Behaviour at

Invariant points

x y (x, y)

-3 ( ) ( )

-2 ( ) ( )

-1 ( ) ( )

- ( ) ( )

- ( ) ( )

0 ( ) ( )

( ) ( )

( ) ( )

1 ( ) ( )

2 ( ) ( )

3 ( ) ( )



1. Graph the original function. 2. Asymptotes use a dashed line to sketch the vertical and horizontal asymptotes.

3. Points use points (x, y) on original function to graph points (x, ) on reciprocal

function. 4. Join connect your points with a smooth curve and predict the behaviour of the graph

using the asymptotes as a guide.

generalize to examine some reciprocal properties:

Original Function: Reciprocal Function:

The point (a, b) on the original function.

The x-intercepts of the original function

Point(s) on the original function where y = 1

Point(s) on the original function where y = 1

The original function has a small y-value.

The original function has a large y-value

The function increases as you move left to right

The function decreases as you move left to right

How do we go about graphing RECIPROCAL functions without using a table of values?!?

You must be able to graph linear functions in the form and quadratic

functions in the form and



Example 2:

a) Graph the reciprocal function

b) Determine the equation of any asymptotes.

c) Determine the domain and range of

the function.

Example 3: Given ;

a) Graph the functions

b) Determine the equation of any asymptotes.

c) Determine the domain

and range of

Assignment Page 403, #2ab, 3ab, 5ab, 7acd, 10a





1. Graph the original function. 2. Asymptotes use a dashed line to sketch the vertical and horizontal asymptotes.

3. Points use points (x, y) on original function to graph points (x, ) on reciprocal

function. 4. Join connect your points with a smooth curve and predict the behaviour of the graph using the asymptotes as a guide.

7.4 Reciprocal Functions: Part 2

The steps for graphing the reciprocal of a quadratic functions are the same as graphing the reciprocal of

Let us examine the graph of the reciprocal function of the most basic quadratic function,

Example 1:

a) Graph the function

b) Determine the equation of any

asymptotes.

b) Determine the domain and range

the function.

As we can see from the above function, there is one vertical asymptote. The number of zeros of the

original quadratic function will determine the number of vertical asymptotes in the reciprocal function.

Use the characteristic regarding the increasing/decreasing value of the function and its reciprocal to

predict behavior of the reciprocal function.



Example 2:



asymptotes.


the function.

Example 3:



asymptotes.


the function.



Example 4:



asymptotes.


the function.

Assignment Page 403, #2cd, 3cd, 5cd, 8bcd, 9, 10b, 15





Ch. 7 Review

Key Ideas Absolute Value / Piecewise Notation Absolute Value Functions Solving Absolute Equations

o Graphically o Algebraically

Reciprocal Functions o Vertical and Horizontal Asyptotes o End behavior o Domain and Range

Practice 1) The zero(es) of 21 occurs when

a) 21

x b) 21

x c) 2x d)

2) Determine the zeros of function .

a) 101x b) c) d)

1) Which of the following is the graph of .

a)

b)

c)

d)



2) Given a linear function, , which of the following is the graph of the reciprocal function,

)(1xf

y ?

a)

b)

c)

d)

3) Determine the equations of the vertical asymptotes of 9

12x

y .

a) b) 3x c) only d) No V.A.

4) Evaluate

5) Evaluate if .

6) Write as a piece-wise function. The graph of )(xfy is shown below.



7) Write |)3)(1(|)( xxxf as a piece-wise function.

8) Write |13|)( 2 xxxf as a piece-wise function.

9) Graph each absolute value function

a)

b) |3)2(| 2xy



10) Solve 7|32| x . Case 1: Case 2:

11) Solve .

Case 1: Case 2:



12) Solve

Case 1: Case 2:

13) Solve by graphing 3|32| 2 xxx

Solution(s): ____________________



14) Given the function of )(xfy ,

i. Sketch the graph of )(

1xf

y . Clearly show invariant points and a few other points.

ii. Draw and label the asymptote(s) in dotted line(s).

iii. State the Domain and range of )(

1xf

y

a) b)



c)

b)

Assignment Page 410, #1-3, 5, 6-8, 11-12, 15-16. Worksheet. # 18



Ch. 7 Review Worksheet

18. Given the function

i. Graph the function

ii. State the equations of any asymptotes

iii. Determine the domain and range of

a)

b)



c)

d)













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Ch 7 Notes - WordPress.com · Ch. 7: Absolute Value and Reciprocal Functions Loo/Lee/Ko Page 26 of 41 Example 2: a) Graph the reciprocal function b) Determine the equation of any