Pre-Calculus - NJCTLcontent.njctl.org/courses/math/pre-calculus/parent...2015/03/24 · Pre-Calculus Parent Functions 2015-03-23 Slide 3 / 305 Table of Contents Linear Functions Exponential

Slide 1 / 305 Slide 2 / 305

Pre-Calculus

Parent Functions

www.njctl.org

2015-03-23

Slide 3 / 305

Table of Contents

Linear Functions

Exponential FunctionsLogarithmic FunctionsProperties of LogsCommon Logse and lnGrowth and DecayLogistic FunctionsTrig FunctionsPower Functions

Positive Integer PowersNegative Integer PowersRational Powers

Rational Functions

click on the topic to go to that section

Step, Absolute Value, Identity and Constant Functions

Slide 4 / 305

Linear Functions

Return toTable of

Contents

Slide 5 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

1 What is the y-intercept of this line?

Linear Functions

Teac

her

Teac

her

Slide 6 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

2 What is the x-intercept of this line?

Linear Functions

Teac

her

Teac

her

http://www.njctl.org




page2svg

page3svg

page4svg

page159svg

page160svg

page161svg

page210svg

page199svg

page204svg

page268svg

page269svg

page278svg

page287svg

page294svg

page120svg

page1svg

Slide 7 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

An infinite number of lines can pass through the same location on the y-axis...they all have the same y-intercept.

Examples of lines with a y-intercept of ____ are shown on this graph. What's the difference between them (other than their color)?

Consider this...

Linear Functions

Slide 8 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

The red line has a positive slope, since the line rises from left to the right.

The Slope of a Linerun

rise

Linear Functions

Slide 9 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

The orange line has a negative slope, since the line falls down from left to the right.

The Slope of a Line

riserun

Linear Functions

Slide 10 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

The purple line has a slope of zero, since it doesn't rise at all as you go from left to right on the x-axis.

The Slope of a LineLinear Functions

Slide 11 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

The Slope of a Line

The black line is a vertical line. It has an undefined slope, since it doesn't run at all as you go from the bottom to the top on the y-axis.

rise0

= undefined

Linear Functions

Slide 12 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

While we can quickly see if the slope of a line is positive, negative or zero...we also need to determine how much slope it has...we have to measure the slope of a line.

Measuring the Slope of a Line

rise

run

Linear Functions

Slide 13 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

The slope of the line is just the ratio of its rise over its run.

The symbol for slope is "m".

So the formula for slope is:


rise

run

slope = riserun

Linear Functions

Slide 14 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

The slope is the same anywhere on a line, so it can be measured anywhere on the line.


rise

run

slope = riserun

Linear Functions

Slide 15 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

For instance, in this case we measure the slope by using a run from x = 0 to x = +6: a run of 6.

During that run, the line rises from y = 0 to y = 8: a rise of 8.


rise

run

slope = riserun

m = 86

m = 43

Linear Functions

Slide 16 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

But we get the same result with a run from x = 0 to x = +3: a run of 3.



riserun

slope = riserun

m = 43

Linear Functions

Slide 17 / 305

But we can also start at x = 3 and run to x = 6 : a run of 3.



rise

run

slope = riserun

m = 43

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Slide 18 / 305

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

But we can also start at x = -6 and run to x = 0: a run of 6.

During that run, the line rises from y = -8 to y = 0: a rise of 8.


rise

run

slope = riserun

m = 86

m = 43

Linear Functions

Slide 19 / 305

Slope formula can be used to find the constant of change in a "real world" problem.

When traveling on the highway, drivers will set the cruise control and travel at a constant speed this means that the distance traveled is a constant increase.

The graph at the right represents such a trip. The car passed mile-marker 60 at 1 hour and mile-marker 180 at 3 hours. Find the slope of the line and what it represents.

m= = =

So the slope of the line is 60 and the rate of change of the car is 60 miles per hour.

180 miles-60 miles 3 hours-1 hours

120 miles 2 hours

60 miles hour

Time(hours)

Distance(miles)

(1,60)

(3,180)

Linear Functions

Slide 20 / 305

If a car passes mile-marker 100 in 2 hours and mile-marker 200 in 4 hours, how many miles per hour is the car traveling?

Use the information to write ordered pairs (2,100) and (4,200).

Linear Functions

Teac

her

Teac

her

Slide 21 / 305

3 If a car passes mile-marker 90 in 1.5 hours and mile-marker 150 in 3.5 hours, how many miles per hour is the car traveling?

Linear FunctionsTe

ache

rTe

ache

r

Slide 22 / 305

4 How many meters per second is a person running if they are at 10 meters in 3 seconds and 100 meters in 15 seconds?

Linear Functions

Teac

her

Teac

her

Slide 23 / 305

5 Which equation does the graph represent?

A y = -6x-6B y = -1x+6C y = 6x+6D y = x-6

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Teac

her

Teac

her

Slide 24 / 305


A y = 4xB y = x+4C y = 4D y = x

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Teac

her

Teac

her

Slide 25 / 305


A y = -x + 6B y = -3x+6C y = -3x+3D y = 6 2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Teac

her

Teac

her

Slide 26 / 305


A y = 2x-6B y = -6x+3C y = 4x-3D y = -3x+6 2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Teac

her

Teac

her

Slide 27 / 305


A y = 10x+10B y = -2x+10C y = 5x+10D y = -5x+10

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Teac

her

Teac

her

Slide 28 / 305

10 Which graph represents the equation y = 3x-2?

A line AB line BC line CD line D 2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

AB

C

D

Linear Functions

Teac

her

Teac

her

Slide 29 / 305

11 Which graph represents y = -1/2x+3?


4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

AB

C

D

Linear Functions

Teac

her

Teac

her

Slide 30 / 305

A line can be graphed by using the x- and y- intercepts.

The technique of using intercepts works well when an equation is written in STANDARD FORM. Standard form looks like

Ax + By = C, where A, B, and C are integers and A>0.and the Greatest Common Factor of A, B, and C is 1.

Linear Functions

Slide 31 / 305

12 Is 4x -3y = 6 in standard form?

Yes

No

Linear Functions

Teac

her

Teac

her

Slide 32 / 305

13 Is -2x + 3y = 7 in standard form?

Yes

No

Linear Functions

Teac

her

Teac

her

Slide 33 / 305

14 Is 4x - 8y = 6 in standard form?

Yes

No

Linear FunctionsTe

ache

rTe

ache

r

Slide 34 / 305

15 The standard form of y = -2x +7 is

A 2x - y = 7B 2x + y = 7

C -2x - y = 7

D -2x + y = -7

Linear Functions

Teac

her

Teac

her

Slide 35 / 305

16 The standard form of y = 3x -7 is

A 3x - y = 7B 3x + y = -7C -3x - y = 7

D -3x + y = -7

Linear Functions

Teac

her

Teac

her

Slide 36 / 305

17 The standard form of y = -2/3x + 5 is

A 2x - 3y = 15B 2x + 3y = 15

C -2x - 3y = 15

D -2x + 3y = 15

Linear Functions

Teac

her

Teac

her

Slide 37 / 305

18 The standard form of y - 4 = 1/2(x +7) is

A x - 2y = -15B x + 2y = 15

C x - 2y = 15D x + 2y = -15

Linear Functions

Teac

her

Teac

her

Slide 38 / 305

Given the equation 4x-3y=12

Linear Functions

Find the x-intercept.

x-intercept: Let y=0: 4x-3(0)=12 4x=12 x=3 so x-intercept is (3,0)

Find the y-intercept.

y-intercept: Let x=0: 4(0)-3y=12 -3y=12 y=-4 so y-intercept is (0-4)

Slide 39 / 305

Given the equation 4x-3y=12 we found the x-intercept is (3,0) and the y-intercept is (0,-4).

Graph the intercepts and then the line that passess through them.

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Linear Functions

Slide 40 / 305

What are the x- and y- intercepts of y=3x-9?

Linear Functions

Teac

her

Teac

her

Slide 41 / 305

19 Given the equation y = 1/2x-7, what is x when y=0?

Linear Functions

Teac

her

Teac

her

Slide 42 / 305

20 Given the equation y = 1/2x-7, what is y when x=0?

Linear Functions

Teac

her

Teac

her

Slide 43 / 305

21 Given the equation y-4 = 4(x+2), what is x when y=0?

Linear Functions

Teac

her

Teac

her

Slide 44 / 305

22 Given the equation y-4 = 4(x+2) what is y when x=0?

Linear Functions

Teac

her

Teac

her

Slide 45 / 305

Horizontal and Vertical Lines

Return to Table of Contents

Slide 46 / 305

A VERTICAL line goes "up and down".It has the equation x=a number,the x-intercept. Notice no y in the equation.

An Example of a Vertical Line x=3

Horizontal liney=2

A HORIZONTAL line goes "sideways".It has the equation y=a number,the y-intercept. Notice no x in the equation.

Linear Functions

Slide 47 / 305

An Example of a Vertical Line x=3

Horizontal liney=2

A horizontal line has a slope of 0,as opposed to a vertical line which has an undefined slope.

2 points on the horizontal line are (0,2) and (5,2): m= = =0

2 points on the vertical line are (3,4) and (3,9): m= = =undefined because you can't divide by 0.

2-20-5

0-5

4-93-3

-5 0

Linear Functions

Slide 48 / 305

23 Is the following equation that of a vertical line,a horizontal line, neither, or cannot be determined: y=4

A VerticalB HorizontalC NeitherD Cannot be determined

Linear Functions

Teac

her

Teac

her

page1svg

Slide 49 / 305

24 Is the following equation that of a vertical line, a horizontal line, neither, or cannot determine: x+2y = 9

A VerticalB HorizontalC NeitherD Cannot be Determined

Linear Functions

Teac

her

Teac

her

Slide 50 / 305

25 Is the following line that of a vertical, a horizontal, neither, or cannot be determined: x = -23


Linear Functions

Teac

her

Teac

her

Slide 51 / 305

26 Is the following equation that of a vertical line, a horizontal line,neither, or cannot be determined: 2x-3 = 0


Linear FunctionsTe

ache

rTe

ache

r

Slide 52 / 305

27 The intercepts method of graphing could not have been used to graph which of the following graphs? There is more than 1 answer.

A

B

C

D

E

F

G

H

A

B

C

D

E

F

G

H

Linear Functions

Teac

her

Teac

her

Slide 53 / 305

28 Which of the following equations can't be graphed using the intercepts method? There are multiple answers.A y=3B y-2 = 1/2(x+9)C y = -3xD x = -4E y = 4x+7

F 3x - 4y =12G x = 2y -8H y=x

I 2y = 8x-6

J y+4 = (x+1)

Linear Functions

Teac

her

Teac

her

Slide 54 / 305

Parallel and Perpendicular Lines


page1svg

Slide 55 / 305

The lines at the right are parallel lines. Notice that their slopes are all the same.

Parallel lines all have the slopes because if they change at different rates eventually they would intersect.

This also works for vertical and horizontal lines.

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

h(x)=x+6

q(x)=x+2r(x)=x-1 s(x)=x-5

Linear Functions

Slide 56 / 305

29 Which line is parallel to y = -3x-17?

A y = -3x+1B y = 1/3xC y = 5D x = 2

Linear Functions

Teac

her

Teac

her

Slide 57 / 305

30 Which line is parallel to y = 0?

A y = -3x+1B y = 1/3xC y = 5D x = 2

Linear Functions

Teac

her

Teac

her

Slide 58 / 305

31 Which line is parallel to y + 1 = -3(x-6)?

A y = -3x+1B y = 1/3xC y = 5D x = 2

Linear Functions

Teac

her

Teac

her

Slide 59 / 305

In the diagram the 2 lines form a right angle, when this happenslines are said to perpendicular.

Look at their slopes. This time theyare not the same instead they areopposite reciprocals and -3x 1/3 = -1.

h(x)=-3x-11

g(x)=1/3x-2

Linear Functions

Slide 60 / 305

A) y=4x-2 is perpendicular to

B) y=-1/5x+1 is perpendicular to

C) y-2=-1/4(x-3) is perpendicular to

D) 5x-y=8 is perpendicular to

E) y=1/6x is perpendicular to

F) y-9=-5(x-.4) is perpendicular to

G) y=-6(x+2) is perpendicular to

Perpendicular Equation Bank

y=1/6x-6

y=-1/4x-3y=4x+1

6x+y=10

1/5y=x-2

y=-1/5x+9

y=1/5x

(Drag the equationto complete the

statement.)

Linear Functions

Teac

her

Teac

her

Slide 61 / 305

The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines.

Why?

Linear Functions

Teac

her

Teac

her

Slide 62 / 305

32 Which line is perpendicular to y = -3x+2 ?

A y = -3x+1B y = 1/3xC y = 5D x = 2

Linear Functions

Teac

her

Teac

her

Slide 63 / 305

33 Which line is perpendicular to y+2 = -3(x-6) ?

A y = -3x+1B y = 1/3xC y = 5D x = 2

Linear FunctionsTe

ache

rTe

ache

r

Slide 64 / 305

34 Which line is perpendicular to y = 0 ?

A y = -3x+1B y = 1/3xC y = 5D x = 2

Linear Functions

Teac

her

Teac

her

Slide 65 / 305

Graphing Using the Point-Slope

Equation of a Line


Slide 66 / 305

The point-slope equation for a line isy - y1 = m (x - x1)where m is the slope and(x1,y1) is a point on the line.

(x1,y1) can be graphed and then applying m, a second point can be graphed. The line containing all of the (x,y) can be graphed.

1) Plot (x1,y1)2) From that point, use the slope (m) to plot a second point.3)Graph the line through these 2 points.

This line represents all of the points that are solutions to the equation.

Linear Functions

page1svg

Slide 67 / 305

The point-slope equation for a line isy - y1 = m (x - x1)where m is the slope and(x1,y1) is a point on the line.

Notice that the signs of (x1,y1) arechanged from the formula.

Given the equation y - 3 = 2 (x + 7)the line passes through (-7,3) and has a slope of 2.

Now the graph can be drawn.

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

21

(-7,3)

Linear Functions

Slide 68 / 305

35 What is the slope of y-3 = 4(x+2)?

Linear Functions

Teac

her

Teac

her

Slide 69 / 305

36 Which line represents y+5 = -1/3(x-4)?

A line AB line BC line CD line D

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

AB

CD

Linear FunctionsTe

ache

rTe

ache

r

Slide 70 / 305

37 Which is the slope and a point on the line y-1 = 1/3(x)?

A m=1/3; (-1,0)B m= -1/3; (0,-1)C m=1/3; (0,1)D m is undefined; (0,1)

Linear Functions

Teac

her

Teac

her

Slide 71 / 305

Given the equation y = 5(x -1)Determine the point on the line and the slope.

Now graph the line.

Linear Functions

Teac

her

Teac

her

Slide 72 / 305

Given the equation y -1 = 2/5 (x +5)Determine the point on the line and the slope.

Graph the line representing the equation.

Linear Functions

Teac

her

Teac

her

Slide 73 / 305

38 What is the slope and a point on the line y+5 = -3(x-4)?

A m=-3; (4,-5)B m=-3; (-4,5)C m=3; (4,-5)D m=3; (-4,5)

Linear Functions

Teac

her

Teac

her

Slide 74 / 305

39 Which line represents y+6 = -3(x-4)?


4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

A

B

C

D

Linear Functions

Teac

her

Teac

her

Slide 75 / 305

40 Which line represents y+5 = -3(x-4)?


4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

A B

C

D

Linear Functions

Teac

her

Teac

her

Slide 76 / 305

41 Which point is on the line y-3 = 4(x+2)?

A (-3,2)B (3,-2)C (2,-3)D (-2,3)

Linear Functions

Teac

her

Teac

her

Slide 77 / 305

Given the equation y +4 = 1/3 (x +2)Determine the point on the line and the slope.

Graph the line representing the equation.

Linear Functions

Teac

her

Teac

her

Slide 78 / 305

To write an equation in point-slope form: First find the slope: -the slope can be given:for example "the slope is"or"m=" -given two points, use the slope formula: m= -given a parallel line, use the same slope -given a perpendicular, use the opposite reciprocal slope After finding the slope use a point on the line to write the equation.

If the directions ask for a different form, like y=mx+b, convert point slope into the desired form.

y2-y1

x2 -x1

Linear Functions

Slide 79 / 305

Write the equation of the line with slope of 1/2 and through the point (2,5).

Linear Functions

Teac

her

Teac

her

Slide 80 / 305

Write the equation of the line with slope -3 and containing the point (1,2) in slope-y-intercept form.

Linear Functions

Teac

her

Teac

her

Slide 81 / 305

Write the equation of the line through(5,6) and (7,1).

Linear Functions

Teac

her

Teac

her

Slide 82 / 305

Write the equation of the line through (3,1) and (4,5) in standard form.

Linear Functions

Teac

her

Teac

her

Slide 83 / 305

Write the equation of the line with x-intercept of 5 and y-intercept of 10.

Linear Functions

Teac

her

Teac

her

Slide 84 / 305

Write the equation of the line through (4,1) and parallel to the line y=3x-6.

Linear Functions

Teac

her

Teac

her

Slide 85 / 305

Write the equation of the line through (-3,-2) and perpendicular to y=-4/5x+1 in slope-y-intercept form.

Linear Functions

Teac

her

Teac

her

Slide 86 / 305

42 Which is the equation of the line with slope of -3 and through (1,-4)?

A y-4 = -3(x+1)B y+4 = 3(x-1)C y+4 = -3(x-1)D y+4 = -3(x+1)

Linear Functions

Teac

her

Teac

her

Slide 87 / 305

43 Which is the equation of the line through (1,3) and (2,5) in slope y-intercept form?

A y-3 = 2(x-1)B y = 2x+1C y-5 = 2(x-2)D y = 2x+3

Linear FunctionsTe

ache

rTe

ache

r

Slide 88 / 305

44 Which is the equation of the line through (-2,5) and perpendicular to y-7 = -1/3(x+9) ?

A y-5 = -1/3(x+2)B y-5 = -3(x+2)C y-5 = 3(x-2)D y-5 = 3(x+2)

Linear Functions

Teac

her

Teac

her

Slide 89 / 305

Step, Absolute Value, Identity, and Constant

Functions

Return toTable of Contents

Slide 90 / 305

There are special functions that have there own names and graphs.

Constant Function Identity Function

y = bDomain: RealsRange: Reals

y = xDomain: RealsRange: Reals

Linear Functions

page1svg

Slide 91 / 305

Absolute Value Function

y = a|cx -h| + kDomain: RealsRange: -2 < y

Linear Functions

Slide 92 / 305

To Graph an Absolute Value Graph1) Set the value inside of the absolute value sign equal to zero and solve.This is x-value of the vertex of the graph.

2) Create a table. Use the solution to step one as a middle value by picking a couple of points smaller and a couple larger. Complete the table.

3) Graph the points.

4) Connect the points.

5) As a check, if the number in front of the absolute value sign is positive the "V" opens up, if its negative it opens down.

X Y

1 0

2 -2

3 -4

4 -2

5 0

D:Reals; R: -4 < y

Linear Functions

Slide 93 / 305

Graph y = -3| 2x + 4| + 5

X Y

D: _______ ; R:__________

Linear Functions

Teac

her

Teac

her

Slide 94 / 305

Graph y = 2| -2x + 6| - 2

X Y

D: _______ ; R:__________

Linear Functions

Teac

her

Teac

her

Slide 95 / 305

45 What is the x value of the vertex of y = | 2x -1| +2

A -2B 1

C 1/2

D 2

Linear Functions

Teac

her

Teac

her

Slide 96 / 305

46 Which of the following is the correct graph of y = | x+4| - 2 ?

A B

C D

Linear Functions

Teac

her

Teac

her

Slide 97 / 305

47 Which of the following is the correct graph of y = | x - 4| - 2 ?

A B

C D

Linear Functions

Teac

her

Teac

her

Slide 98 / 305

48 Which of the following is the correct graph of y = -2 | 3x + 9| + 5 ?

A B

C D

Linear Functions

Teac

her

Teac

her

Slide 99 / 305

49 Graph y = |2x - 6| + 3

Linear FunctionsTe

ache

rTe

ache

r

Slide 100 / 305

50 What is the domain of the graphed function?

A Set of Integers

B Set of Reals

C x > -3D x < -3

Linear Functions

Teac

her

Teac

her

Slide 101 / 305

51 What is the range of the graphed function?

A Set of Integers

B Set of Reals

C y > -3D y < -3

Linear Functions

Teac

her

Teac

her

Slide 102 / 305


A Set of Integers

B Set of Reals

C x > 3

D x < 3

Linear Functions

Teac

her

Teac

her

Slide 103 / 305


A Set of Integers

B Set of Reals

C y > 3

D y < 3

Linear Functions

Teac

her

Teac

her

Slide 104 / 305

Greatest Integer Functions

[2] = 2[2.1] = 2[2.3] = 2[2.5] = 2[2.75] = 2[2.999] = 2[3] = 3

[-2] = -2[-2.1] = -3[-2.3] = -3[-2.5] = -3[-2.75] = -3[-2.999] = -3[-3] = -3

The [ ] tell you to round to the preceding integer. Think round to the left on a number line.

[ ] are a grouping sign and the inside should be simplified before rounding.

Linear Functions

Slide 105 / 305

Evaluate

[3.5 + .6]

[3.7 - .8] [ 2 - 2.1]

3[2.4 +.2] [3(2.4) + .2] 3[2.4] + .2 4[2.1 - 2]2

Linear FunctionsTe

ache

rTe

ache

r

Slide 106 / 305

54 Evaluate [2.6]Linear Functions

Teac

her

Teac

her

Slide 107 / 305

55 Evaluate [5+2]Linear Functions

Teac

her

Teac

her

Slide 108 / 305

56 Evaluate [ -2.6 ]Linear Functions

Teac

her

Teac

her

Slide 109 / 305

57 Evaluate [ -2.1 ]Linear Functions

Teac

her

Teac

her

Slide 110 / 305

58 Evaluate 3[2.6 + .5]2Linear Functions

Teac

her

Teac

her

Slide 111 / 305

Graphing a Greatest Integer Function

It also called a StepFunction because ofthe shape of its graph.

Domain: RealsRange: Integers

Linear Functions

Slide 112 / 305

Graphing a Greatest Integer Function

1) Find the values of x that don't have to be rounded. The inside of [ ] determines that.

2) Make a table. Pick values around the integer values in step 1.Remember our graph will look like steps so once we know the height and width of each step we can repeat the pattern.

3) Graph. Continue the pattern to complete.

X Y

0 0

0.2 0

0.4 0

0.5 1

0.8 1

0.9 1

1 2

1.5 3(graph is on next page)

Linear Functions

Slide 113 / 305

X Y

0 0

0.2 0

0.4 0

0.5 1

0.8 1

0.9 1

1 2

1.5 3

Linear Functions

Slide 114 / 305

Graph y = [x +1]

X Y

Linear Functions

Slide 115 / 305

Graph y = [4x]

X Y

Linear Functions

Slide 116 / 305

Graph y = 2[x -3]

X Y

Linear Functions

Slide 117 / 305

Graph y = [.5x]

X Y

Linear Functions

Slide 118 / 305

Graph y = 2[.5x + 1]

X Y

Linear Functions

Slide 119 / 305


A Set of Integers

B Set of Reals

C Set of Odd Integers

D Set of Even Integers

Linear Functions

Teac

her

Teac

her

Slide 120 / 305


A Set of Integers

B Set of Reals



Linear Functions

Teac

her

Teac

her

Slide 121 / 305


A Set of Integers

B Set of Reals



Linear Functions

Teac

her

Teac

her

Slide 122 / 305


A Set of Integers

B Set of Reals



Linear Functions

Teac

her

Teac

her

Slide 123 / 305

Exponential Functions

Return toTable of

Contents

Slide 124 / 305

We have looked at linear growth, where the amount of change is constant.

X Y

1 3

2 5

3 7

4 9

If x = 5 what is y?


Teac

her

Teac

her

Slide 125 / 305

When the rate of growth increases as time passes, the function is said to be exponential.

X Y

1 1

2 2

3 4

4 8

If x = 5 what is y?


Teac

her

Teac

her

Slide 126 / 305

We will also looking at exponential decay. Think of it as you 8 m&m's and each day you eat half.

How many will be left of the 5th day?

X Y

0 8

1 4

2 2

3 1

4 0.5


Teac

her

Teac

her

page1svg

Slide 127 / 305

How can we recognize an Exponential function?

Slide 128 / 305

From a GraphThe exponential function has a curved shape to it.

D: {reals} R: {positive reals}

Exponential Growth Exponential Decay


Slide 129 / 305

63 Which of these are exponential growth graphs?A B C D E

F G H I


Teac

her

Teac

her

Slide 130 / 305

64 Which of these are exponential decay graphs?A B C D E

F G H I


Teac

her

Teac

her

Slide 131 / 305

The general form of an exponential is

where x is the variable and a, b, and c are constants.

b is the growth rate. If b > 1 then its exponential growth If 0< b < 1 then its exponential decay

c is the horizontal asymptote

a + c is the y-intercept


Slide 132 / 305

65 Consider the following equation, is it exponential growth or decay?

A growthB decay


Teac

her

Teac

her

Slide 133 / 305

66 Considering the following equation, what is the equation of the horizontal asymptote?

A y=2B y=3C y=4D y=5


Teac

her

Teac

her

Slide 134 / 305

67 Considering the following equation, what is the y-intercept?

A (0,3)

B (0,4)C (0,7)D (0,9)


Teac

her

Teac

her

Slide 135 / 305

68 Consider the following equation is it exponential growth or decay?

A growthB decay


Teac

her

Teac

her

Slide 136 / 305


A y=.2B y=1C y=3D y=4


Teac

her

Teac

her

Slide 137 / 305


A (0,.2)B (0,1)C (0,3)D (0,4)


Teac

her

Teac

her

Slide 138 / 305

71 Consider the following equation is it exponential growth or decay?

A growthB decay


Teac

her

Teac

her

Slide 139 / 305


A y=0B y=1C y=3D y=4


Teac

her

Teac

her

Slide 140 / 305


A (0,0)B (0,1)C (0,3)D (0,4)


Teac

her

Teac

her

Slide 141 / 305

Sketching the graph of an exponential requires using a, b, and c.

1) Identify horizontal asymptote (y = c)2) Determine if graph is decay or growth3) Graph y-intercept (0,a+c)4) Sketch graph

Example:

Step 1 Step 2 Step 3 Step 4

y = 2growth (0,5)


Slide 142 / 305

Teac

her

Teac

her

Graph


Slide 143 / 305

Logarithmic Functions

Return toTable of

Contents

Slide 144 / 305

Logarithm functions are the inverses of exponential functions.

Logs have the same domain as the exponential had range,that is a you cannot take the log of 0 or a negative.

Exponential Log


page1svg

Slide 145 / 305 Slide 146 / 305

Rewrite the following in exponential form.


Teac

her

Teac

her

Slide 147 / 305 Slide 148 / 305

75 Which of the following is the correct logarithmic form of ?

A

B

C

D


Teac

her

Teac

her

Slide 149 / 305

76 Which of the following is the correct exponential form of ?

A

B

C

D


Teac

her

Teac

her

Slide 150 / 305

77 Which of the following is the correct exponential form of ?

A

B

C

D


Teac

her

Teac

her

Slide 151 / 305 Slide 152 / 305

78 Solve


Teac

her

Teac

her

Slide 153 / 305

79 Solve

Logarithmic FunctionsTe

ache

rTe

ache

r

Slide 154 / 305

Slide 155 / 305 Slide 156 / 305

Properties of Logs

Return toTable of

Contents

page1svg

Slide 157 / 305

Properties of Logs

These rules may seem strange but recall that logsare a way of dealing with exponents, so when we multiplied like bases we added the exponents. Just likerule 1.

Properties of Logs

Slide 158 / 305

Examples: Use the Properties of Logs to expand

Properties of Logs

Teac

her

Teac

her

Slide 159 / 305

82 Which choice is the expanded form of the following

A

B

C

D

Properties of LogsTe

ache

rTe

ache

r

Slide 160 / 305


A

B

C

D

Properties of Logs

Teac

her

Teac

her

Slide 161 / 305


A

B

C

D

Properties of Logs

Teac

her

Teac

her

Slide 162 / 305


A

B

C

D

Properties of Logs

Teac

her

Teac

her

Slide 163 / 305 Slide 164 / 305

Slide 165 / 305 Slide 166 / 305

Slide 167 / 305 Slide 168 / 305

Slide 169 / 305 Slide 170 / 305

Slide 171 / 305 Slide 172 / 305

Slide 173 / 305 Slide 174 / 305

93 Solve the following equation:

Properties of Logs

Teac

her

Teac

her

Slide 175 / 305 Slide 176 / 305

Slide 177 / 305


Properties of Logs

Teac

her

Teac

her

Slide 178 / 305

Slide 179 / 305


Properties of Logs

Teac

her

Teac

her

Slide 180 / 305

Common Logs

Return toTable of

Contents

page1svg

Slide 181 / 305 Slide 182 / 305

Solving Exponential Equations Using Common Logs

We can solve the variable as exponent using common logs

Common Logs

Slide 183 / 305 Slide 184 / 305

100 Solve the following equation.

Common Logs

Teac

her

Teac

her

Slide 185 / 305


Common Logs

Teac

her

Teac

her

Slide 186 / 305


Common Logs

Teac

her

Teac

her

Slide 187 / 305 Slide 188 / 305

Slide 189 / 305 Slide 190 / 305

Slide 191 / 305

e and ln

Return toTable of

Contents

Slide 192 / 305

e has a constant value of about 2.71828

e is the number used when something is continually growing, like a bacteria colony or an oil spill.

The Natural Log is the inverse function of a base e function.

e and ln

page1svg

Slide 193 / 305

Work with e and ln the same way you did 10 and log.

For example:

e and ln

Slide 194 / 305

Write the following in the equivalent exponential or log form.

e and ln

Teac

her

Teac

her

Slide 195 / 305 Slide 196 / 305

Slide 197 / 305

The amount of money in a savings account, A, can be found using the continually compounded interest formula of

A=Pert , where P is the principal (amount deposited), r is the annual interest rate (in decimal form),and t is time in years.

If $500 is invested at 4% for 2 years, what will account balance be?

e and ln

Teac

her

Teac

her

Slide 198 / 305

Again,using the compounded continually formula of A=Pert If $500 is invested at 4% , how long until the account balance is doubled?

e and ln

Teac

her

Teac

her

Slide 199 / 305 Slide 200 / 305

107 Find the value of x.

e and ln

Teac

her

Teac

her

Slide 201 / 305

108 Find the value of x.e and ln

Teac

her

Teac

her

Slide 202 / 305

Slide 203 / 305 Slide 204 / 305

111 Find the value of x.

e and ln

Teac

her

Teac

her

Slide 205 / 305

112 The amount of money in a savings account, A, can be found using the continually compounded interest formula of A=Pert , where P is the principal (amount deposited), r is the annual interest rate (in decimal form),and t is time in years.

If $1000 is invested at 4% for 3 years,what is the account balance?

e and ln

Teac

her

Teac

her

Slide 206 / 305

113 The amount of money in a savings account, A, can be found using the continually compounded interest formula of A=Pert , where P is the principal (amount deposited), r is the annual interest rate (in decimal form), and t is time in years.

If $1000 is invested at 4%, how long until the account balance is doubled?

e and ln

Teac

her

Teac

her

Slide 207 / 305

Growth and Decay

Return toTable of

Contents

Slide 208 / 305

Formulas to remember:Simple Interest

Compounded Interest(annually)

Compounded Interest

Compounded Continually (instantaneously)

AbbreviationsI = interestP= principal (deposit)r= interest rate (decimal)t= timen= number of compoundings in one unit of t

Growth & Decay

Slide 209 / 305

Example: A bacteria constantly grows at a rate of 10% per hour, if initially there were 100 how long till there were 1000?

Growth & Decay

Teac

her

Teac

her

Slide 210 / 305

A new car depreciates in value at a rate of 8% per year. If a 5 year old car is worth $20,000,how much was it originally worth?

(Hint: Since the value of the car is going down, the rate is -.08)

Growth & Decay

Teac

her

Teac

her

page1svg

Slide 211 / 305

The local bank pays 4% monthly on its savings account, how long would it take for a deposit, left untouched, to double?

Growth & Decay

Teac

her

Teac

her

Slide 212 / 305

A certain radioactive material has a half-life of 20 years.If 100g were present to start, how much will remain in 7 years?

Use half-life of 20 years to find r.

Growth & Decay

Teac

her

Teac

her

Slide 213 / 305

114 If you need your money to double in 8 years, what must the interest rate be if is compounded continually?

Growth & DecayTe

ache

rTe

ache

r

Slide 214 / 305

115 If you need your money to double in 8 years, what must the interest rate be if is compounded annually?

Growth & Decay

Teac

her

Teac

her

Slide 215 / 305

116 If you need your money to double in 8 years, what must the interest rate be if is compounded quarterly?

Growth & Decay

Teac

her

Teac

her

Slide 216 / 305

117 If an oil spill widens continually at a rate of 15% per hour, how long will it take to go from 2 miles wide to 3 miles wide?

Growth & Decay

Teac

her

Teac

her

Slide 217 / 305

118 NASA calculates that a communications satellite's orbit is decaying exponentially at a rate of 12% per day. If the satellite is 20,000 miles above the Earth. How long until it is visible to the naked eye at 50 miles high, assuming it doesn't burn up on reentry?

Growth & Decay

Teac

her

Teac

her

Slide 218 / 305

119 If the half-life of an element is 50 years, at what rate does it decay?

Growth & Decay

Teac

her

Teac

her

Slide 219 / 305

120 If the half-life of an element is 50 years, how much of the element is left in 10 years?

Growth & Decay

Teac

her

Teac

her

Slide 220 / 305

121 If the half-life of an element is 50 years, how much of the element is left in 15 years?

Growth & Decay

Teac

her

Teac

her

Slide 221 / 305

122 If the half-life of an element is 50 years, how much of the element is lost between years 10 and 15?

Growth & Decay

Teac

her

Teac

her

Slide 222 / 305

Logistic Functions

Return toTable of

Contents

page1svg

Slide 223 / 305

Logistic FunctionsTo visualize a logistic function graph think of how a rumor spreads around the school.

It starts with a couple of people then a few more and continues to spread till everyone has heard a version of it.

The graph would look something like this:

#who

heard

time

Logistic Functions

Teac

her

Teac

her

Slide 224 / 305

Slide 225 / 305

Ex: A new strain of flu shows up at a school of 1000 people one day. The CDC determines that 10 people brought the flu in and that the rate of growth is 20% per day.

Write an equation.

Make a graph.

Identify the point where the spread is increasing the fastest?What happens to the rate of increase after the point in the previous question?

Logistic FunctionsTe

ache

rTe

ache

r

Slide 226 / 305

Slide 227 / 305

Trig Functions

Return toTable of

Contents

Slide 228 / 305

Trigonometric RatiosThe fundamental trig ratios are:

Sine; called "sin" for short

Cosine; called "cos" for short

Tangent; called "tan" for short

Angles are usually named θ: "theta"

So you'll usually see these as:

sinθ; cosθ ; and tanθ

Trig Functions

page1svg

Slide 229 / 305

Trigonometric RatiosThese ratios depend on which angle you are calling θ; never the right angle.

You know that the side opposite the right angle is called the hypotenuse.

The leg opposite θ is called the opposite side.

The leg that touches θ is called the adjacent side.

hypotenuse

adjacent side

θ

opposite side

Trig Functions

Slide 230 / 305

Trigonometric Ratios

There are two possible angles that can be called #.

Once you choose which angle is #, the names of the sides are defined.

You can change later, but then the names of the sides also change.

hypotenuseadjacent side

θ

opposite side

Trig Functions

Slide 231 / 305


sinθ = =opposite sidehypotenuse

opphyp

adjacent sidehypotenuse

adjhypcosθ = =

hypotenuseadjacent side

θ

opposite side

tanθ = opposite sideadjacent side= opp

adj

SOH-CAH-TOA

Trig Functions

Slide 232 / 305

123

3.0 8.5θ

8.0

Trig Functions

Teac

her

Teac

her

Slide 233 / 305 Slide 234 / 305

Slide 235 / 305 Slide 236 / 305

127

7 16θ

14

Trig Functions

Teac

her

Teac

her

Slide 237 / 305 Slide 238 / 305


If you have the sides trig ratios let you find the angles.

But if you have a side and an angle, trig ratios also let you find the other sides.

Trig Functions

Slide 239 / 305

Trigonometric RatiosFor instance, let's find the length of side x.

The side we're looking for is opposite the given angle;

and the given length is the hypotenuse;

so we'll use the trig function that relates these three:

7.0x

30o


opphyp

Trig Functions

Slide 240 / 305


7.0x

30o


opphyp

sinθ = opphyp

opp = (hyp) (sinθ)

x = (7.0)(sin(30o))

x = (7.0)(0.50)

x = 3.5

Trig Functions

Slide 241 / 305

Trigonometric RatiosNow, let's find the length of side x in this case.

The side we're looking for is adjacent the given angle;

and the given length is the hypotenuse;


9.0

x25o


adjhypcosθ = =

Trig Functions

Slide 242 / 305


9.0

x25o

adj = (hyp)(cosθ)

x = (9.0)(cos(25o))

x = (9.0)(0.91)

x = 8.2


adjhypcosθ = =

adjhypcosθ =

Trig Functions

Slide 243 / 305

Trigonometric RatiosNow, let's find the length of side x in this case.

The side we're looking for is adjacent the given angle;

and the given length is the opposite the given angle;


9.0

x

50o


adj

Trig Functions

Slide 244 / 305


9.0

x

50o

opp = (adj)(tanθ)

x = (9.0)(tan(50o))

x = (9.0)(1.2)

x = 10.8


adj

tanθ = oppadj

Trig Functions

Slide 245 / 305 Slide 246 / 305

Slide 247 / 305 Slide 248 / 305

Slide 249 / 305

Graphing a Trig Function

There are two ways to approach graphing trig functions.

1) is to graph cosine and sine in how they relate to each other will yield a circle, which is not a function.

2) is to graph the value of the function for a given theta. will yield a curve that is a function

Graphing is usually done in radians.

Trig Functions

Slide 250 / 305

Slide 251 / 305

Converting between Radians and Degrees

Convert degrees to radians Convert radians to degrees

Trig Functions

Teac

her

Teac

her

Slide 252 / 305

Slide 253 / 305 Slide 254 / 305

Slide 255 / 305

136 Convert radians to degrees:

Trig FunctionsTe

ache

rTe

ache

r

Slide 256 / 305

Slide 257 / 305 Slide 258 / 305

x

cos x

Graphing cos, sin, & tan

Graph by using values from the table .Since the values are based on a circle, values will repeat.

Trig Functions

Slide 259 / 305



sin x

x

Slide 260 / 305



tan x

x

Slide 261 / 305 Slide 262 / 305

Slide 263 / 305 Slide 264 / 305

140 Which choice(s) below satisfy the following condition:

Increasing

A

B

C

D

E

F

G

H

I

J none of the above

Trig Functions

Teac

her

Teac

her

Slide 265 / 305


Concave Up

A

B

C

D

E

F

G

H

I

J none of the above

Trig Functions

Teac

her

Teac

her

Slide 266 / 305


has a relative max

A

B

C

D

E

F

G

H

I

J none of the above

Trig Functions

Teac

her

Teac

her

Slide 267 / 305


Decreasing

A

B

C

D

E

F

G

H

I

J none of the above

Trig FunctionsTe

ache

rTe

ache

r

Slide 268 / 305

Power Functions

Return toTable of

Contents

Slide 269 / 305

Positive Integer Powers

Return toTable of

Contents

Slide 270 / 305

A power function is in the form of:

Where n is a real number

Depending on the value of n we can get a wide variety of graphs.

Power Functions

page1svg

page1svg

Slide 271 / 305

Let's first consider n to be a positive integer.

n is odd n is even

Power Functions

Slide 272 / 305

The end behaviors of positive integer powers of a power

function follow the rules from the last unit.

Power Functions

Slide 273 / 305

144 Identify the function(s) that satisfy the following:

A

B

C

D

E

F

GH

I

J

Power Functions

Teac

her

Teac

her

Slide 274 / 305


AB

C

DE

F

GH

I

J

Power Functions

Teac

her

Teac

her

Slide 275 / 305


AB

C

DE

F

GH

I

J

Power Functions

Teac

her

Teac

her

Slide 276 / 305


AB

C

DE

F

G

H

I

J

always decreasing

Power Functions

Teac

her

Teac

her

Slide 277 / 305

Negative Integer Powers

Return toTable of

Contents

Slide 278 / 305

Now let's consider n to be a negative integer.

Recall what a negative exponent means:

Power Functions

Slide 279 / 305

f(x)=x2g(x)= 1/x2

Compare and contrast:end behaviors and as the function goes to zero

What is f(x)g(x)= ?

Power Functions

Teac

her

Teac

her

Slide 280 / 305

n is evenn is odd

Power Functions

Slide 281 / 305

Power Functions

Slide 282 / 305


A

B

C

D

E

F

G

H

I

J

Power Functions

Teac

her

Teac

her

page1svg

Slide 283 / 305


A

B

C

D

E

F

GH

I

J

Power Functions

Teac

her

Teac

her

Slide 284 / 305


A

B

C

D

E

F

GH

I

J

Power Functions

Teac

her

Teac

her

Slide 285 / 305


A

B

C

D

E

F

GH

I

J

an odd function

Power Functions

Teac

her

Teac

her

Slide 286 / 305

Rational Powers

Return toTable of

Contents

Slide 287 / 305

is defined as the nth root of x

Rational Powers

Slide 288 / 305

Rational Powers

page1svg

Slide 289 / 305

152 Select the choice that satisfies the following:#### the greatest value at x = 10

A

B

C

D

Rational Powers

Teac

her

Teac

her

Slide 290 / 305

153 Select the choice that satisfies the following? the greatest value at x = -10

AB

C

D

Rational Powers

Teac

her

Teac

her

Slide 291 / 305

154 Select the choice that satisfies the following? the least value at x = 10

A

B

C

D

Rational PowersTe

ache

rTe

ache

r

Slide 292 / 305

155 Select the choice that satisfies the following? the least value at x = -10

A

B

C

D

Rational Powers

Teac

her

Teac

her

Slide 293 / 305

Rational Functions

Return toTable of

Contents

Slide 294 / 305

In a previous unit, the end behaviors of rational functions was discussed.

3 casesm>n:m<n:

m=n:

Rational Functions

page1svg

Slide 295 / 305

Since a rational function was defined as:

there may be values of x that make g(x)=0,which would make h(x) undefined

These values of x are discontinuities of h(x).

Rational Functions

Slide 296 / 305

There are 2 kinds of disconitunity:

Removable Essential

Rational Functions

Slide 297 / 305 Slide 298 / 305

Removable Discontinuity

when x = a,

x - a =0 so (x - a) is a factor of both f(x) and g(x).

· factor (x-a) out of both f(x) and g(x)· re-evaluate h(a)· if h(a)=#/0 , then x=a is a vertical asymptote· if h(a)=c , then (a,c) is the location of the hole

Rational Functions

Slide 299 / 305

Examples: graph the following

Rational Functions

Teac

her

Teac

her

Slide 300 / 305

Examples: graph the following

Rational Functions

Teac

her

Teac

her

Slide 301 / 305

Examples: graph the followingRational Functions

Teac

her

Teac

her

Slide 302 / 305

Slide 303 / 305

157 There is a hole at x = c for the equation

Find c.

Rational FunctionsTe

ache

rTe

ache

r

Slide 304 / 305

Slide 305 / 305

Documents

Pre-Calculus - NJCTLcontent.njctl.org/courses/math/pre-calculus/parent...2015/03/24 · Pre-Calculus Parent Functions 2015-03-23 Slide 3 / 305 Table of Contents Linear Functions Exponential