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Pre-Calculus
Parent Functions
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2015-03-23
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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
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Linear Functions
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Contents
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1 What is the y-intercept of this line?
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2 What is the x-intercept of this line?
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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...
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The red line has a positive slope, since the line rises from left to the right.
The Slope of a Linerun
rise
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The orange line has a negative slope, since the line falls down from left to the right.
The Slope of a Line
riserun
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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
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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
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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
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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:
Measuring the Slope of a Line
rise
run
slope = riserun
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The slope is the same anywhere on a line, so it can be measured anywhere on the line.
Measuring the Slope of a Line
rise
run
slope = riserun
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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.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 86
m = 43
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But we get the same result with a run from x = 0 to x = +3: a run of 3.
During that run, the line rises from y = 0 to y = 4: a rise of 4.
Measuring the Slope of a Line
riserun
slope = riserun
m = 43
Linear Functions
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But we can also start at x = 3 and run to x = 6 : a run of 3.
During that run, the line rises from y = 3 to y = 7: a rise of 4.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 43
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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.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 86
m = 43
Linear Functions
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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)
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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).
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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?
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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?
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5 Which equation does the graph represent?
A y = -6x-6B y = -1x+6C y = 6x+6D y = x-6
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6 Which equation does the graph represent?
A y = 4xB y = x+4C y = 4D y = x
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7 Which equation does the graph represent?
A y = -x + 6B y = -3x+6C y = -3x+3D y = 6 2
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8 Which equation does the graph represent?
A y = 2x-6B y = -6x+3C y = 4x-3D y = -3x+6 2
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9 Which equation does the graph represent?
A y = 10x+10B y = -2x+10C y = 5x+10D y = -5x+10
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10 Which graph represents the equation y = 3x-2?
A line AB line BC line CD line D 2
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11 Which graph represents y = -1/2x+3?
A line AB line BC line CD line D 2
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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.
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12 Is 4x -3y = 6 in standard form?
Yes
No
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13 Is -2x + 3y = 7 in standard form?
Yes
No
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14 Is 4x - 8y = 6 in standard form?
Yes
No
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15 The standard form of y = -2x +7 is
A 2x - y = 7B 2x + y = 7
C -2x - y = 7
D -2x + y = -7
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16 The standard form of y = 3x -7 is
A 3x - y = 7B 3x + y = -7C -3x - y = 7
D -3x + y = -7
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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
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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
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Given the equation 4x-3y=12
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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)
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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.
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What are the x- and y- intercepts of y=3x-9?
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19 Given the equation y = 1/2x-7, what is x when y=0?
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20 Given the equation y = 1/2x-7, what is y when x=0?
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21 Given the equation y-4 = 4(x+2), what is x when y=0?
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22 Given the equation y-4 = 4(x+2) what is y when x=0?
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Horizontal and Vertical Lines
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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
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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
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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
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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
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25 Is the following line that of a vertical, a horizontal, neither, or cannot be determined: x = -23
A VerticalB HorizontalC NeitherD Cannot be Determined
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26 Is the following equation that of a vertical line, a horizontal line,neither, or cannot be determined: 2x-3 = 0
A VerticalB HorizontalC NeitherD Cannot be Determined
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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
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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)
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Parallel and Perpendicular Lines
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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.
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q(x)=x+2r(x)=x-1 s(x)=x-5
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29 Which line is parallel to y = -3x-17?
A y = -3x+1B y = 1/3xC y = 5D x = 2
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30 Which line is parallel to y = 0?
A y = -3x+1B y = 1/3xC y = 5D x = 2
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31 Which line is parallel to y + 1 = -3(x-6)?
A y = -3x+1B y = 1/3xC y = 5D x = 2
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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
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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.)
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The rule of using opposite reciprocals will not work for Horizontal and Vertical Lines.
Why?
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32 Which line is perpendicular to y = -3x+2 ?
A y = -3x+1B y = 1/3xC y = 5D x = 2
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33 Which line is perpendicular to y+2 = -3(x-6) ?
A y = -3x+1B y = 1/3xC y = 5D x = 2
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34 Which line is perpendicular to y = 0 ?
A y = -3x+1B y = 1/3xC y = 5D x = 2
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Graphing Using the Point-Slope
Equation of a Line
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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
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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.
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35 What is the slope of y-3 = 4(x+2)?
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36 Which line represents y+5 = -1/3(x-4)?
A line AB line BC line CD line D
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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)
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Given the equation y = 5(x -1)Determine the point on the line and the slope.
Now graph the line.
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Given the equation y -1 = 2/5 (x +5)Determine the point on the line and the slope.
Graph the line representing the equation.
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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)
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39 Which line represents y+6 = -3(x-4)?
A line AB line BC line CD line D 2
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40 Which line represents y+5 = -3(x-4)?
A line AB line BC line CD line D 2
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41 Which point is on the line y-3 = 4(x+2)?
A (-3,2)B (3,-2)C (2,-3)D (-2,3)
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Given the equation y +4 = 1/3 (x +2)Determine the point on the line and the slope.
Graph the line representing the equation.
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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
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Write the equation of the line with slope of 1/2 and through the point (2,5).
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Write the equation of the line with slope -3 and containing the point (1,2) in slope-y-intercept form.
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Write the equation of the line through(5,6) and (7,1).
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Write the equation of the line through (3,1) and (4,5) in standard form.
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Write the equation of the line with x-intercept of 5 and y-intercept of 10.
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Write the equation of the line through (4,1) and parallel to the line y=3x-6.
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Write the equation of the line through (-3,-2) and perpendicular to y=-4/5x+1 in slope-y-intercept form.
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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)
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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
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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)
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Step, Absolute Value, Identity, and Constant
Functions
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There are special functions that have there own names and graphs.
Constant Function Identity Function
y = bDomain: RealsRange: Reals
y = xDomain: RealsRange: Reals
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Absolute Value Function
y = a|cx -h| + kDomain: RealsRange: -2 < y
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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
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Graph y = -3| 2x + 4| + 5
X Y
D: _______ ; R:__________
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Graph y = 2| -2x + 6| - 2
X Y
D: _______ ; R:__________
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45 What is the x value of the vertex of y = | 2x -1| +2
A -2B 1
C 1/2
D 2
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46 Which of the following is the correct graph of y = | x+4| - 2 ?
A B
C D
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47 Which of the following is the correct graph of y = | x - 4| - 2 ?
A B
C D
Linear Functions
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48 Which of the following is the correct graph of y = -2 | 3x + 9| + 5 ?
A B
C D
Linear Functions
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49 Graph y = |2x - 6| + 3
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50 What is the domain of the graphed function?
A Set of Integers
B Set of Reals
C x > -3D x < -3
Linear Functions
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51 What is the range of the graphed function?
A Set of Integers
B Set of Reals
C y > -3D y < -3
Linear Functions
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52 What is the domain of the graphed function?
A Set of Integers
B Set of Reals
C x > 3
D x < 3
Linear Functions
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53 What is the range of the graphed function?
A Set of Integers
B Set of Reals
C y > 3
D y < 3
Linear Functions
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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
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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
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54 Evaluate [2.6]Linear Functions
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55 Evaluate [5+2]Linear Functions
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56 Evaluate [ -2.6 ]Linear Functions
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57 Evaluate [ -2.1 ]Linear Functions
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58 Evaluate 3[2.6 + .5]2Linear Functions
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Graphing a Greatest Integer Function
It also called a StepFunction because ofthe shape of its graph.
Domain: RealsRange: Integers
Linear Functions
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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
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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
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Graph y = [x +1]
X Y
Linear Functions
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Graph y = [4x]
X Y
Linear Functions
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Graph y = 2[x -3]
X Y
Linear Functions
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Graph y = [.5x]
X Y
Linear Functions
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Graph y = 2[.5x + 1]
X Y
Linear Functions
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59 What is the domain of the graphed function?
A Set of Integers
B Set of Reals
C Set of Odd Integers
D Set of Even Integers
Linear Functions
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60 What is the range of the graphed function?
A Set of Integers
B Set of Reals
C Set of Odd Integers
D Set of Even Integers
Linear Functions
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61 What is the domain of the graphed function?
A Set of Integers
B Set of Reals
C Set of Odd Integers
D Set of Even Integers
Linear Functions
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62 What is the range of the graphed function?
A Set of Integers
B Set of Reals
C Set of Odd Integers
D Set of Even Integers
Linear Functions
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Exponential Functions
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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?
Exponential Functions
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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?
Exponential Functions
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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
Exponential Functions
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How can we recognize an Exponential function?
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From a GraphThe exponential function has a curved shape to it.
D: {reals} R: {positive reals}
Exponential Growth Exponential Decay
Exponential Functions
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63 Which of these are exponential growth graphs?A B C D E
F G H I
Exponential Functions
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64 Which of these are exponential decay graphs?A B C D E
F G H I
Exponential Functions
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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
Exponential Functions
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65 Consider the following equation, is it exponential growth or decay?
A growthB decay
Exponential Functions
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66 Considering the following equation, what is the equation of the horizontal asymptote?
A y=2B y=3C y=4D y=5
Exponential Functions
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67 Considering the following equation, what is the y-intercept?
A (0,3)
B (0,4)C (0,7)D (0,9)
Exponential Functions
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68 Consider the following equation is it exponential growth or decay?
A growthB decay
Exponential Functions
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69 Considering the following equation, what is the equation of the horizontal asymptote?
A y=.2B y=1C y=3D y=4
Exponential Functions
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70 Considering the following equation, what is the y-intercept?
A (0,.2)B (0,1)C (0,3)D (0,4)
Exponential Functions
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71 Consider the following equation is it exponential growth or decay?
A growthB decay
Exponential Functions
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72 Considering the following equation, what is the equation of the horizontal asymptote?
A y=0B y=1C y=3D y=4
Exponential Functions
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73 Considering the following equation, what is the y-intercept?
A (0,0)B (0,1)C (0,3)D (0,4)
Exponential Functions
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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)
Exponential Functions
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Graph
Exponential Functions
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Logarithmic Functions
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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
Logarithmic Functions
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Rewrite the following in exponential form.
Logarithmic Functions
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75 Which of the following is the correct logarithmic form of ?
A
B
C
D
Logarithmic Functions
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76 Which of the following is the correct exponential form of ?
A
B
C
D
Logarithmic Functions
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77 Which of the following is the correct exponential form of ?
A
B
C
D
Logarithmic Functions
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78 Solve
Logarithmic Functions
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79 Solve
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Properties of Logs
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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
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Examples: Use the Properties of Logs to expand
Properties of Logs
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82 Which choice is the expanded form of the following
A
B
C
D
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83 Which choice is the expanded form of the following
A
B
C
D
Properties of Logs
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84 Which choice is the expanded form of the following
A
B
C
D
Properties of Logs
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85 Which choice is the expanded form of the following
A
B
C
D
Properties of Logs
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93 Solve the following equation:
Properties of Logs
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96 Solve the following equation:
Properties of Logs
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98 Solve the following equation:
Properties of Logs
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Common Logs
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Solving Exponential Equations Using Common Logs
We can solve the variable as exponent using common logs
Common Logs
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100 Solve the following equation.
Common Logs
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101 Solve the following equation.
Common Logs
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102 Solve the following equation.
Common Logs
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e and ln
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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
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Work with e and ln the same way you did 10 and log.
For example:
e and ln
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Write the following in the equivalent exponential or log form.
e and ln
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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
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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
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107 Find the value of x.
e and ln
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108 Find the value of x.e and ln
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111 Find the value of x.
e and ln
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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
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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
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Growth and Decay
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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
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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
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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
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The local bank pays 4% monthly on its savings account, how long would it take for a deposit, left untouched, to double?
Growth & Decay
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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
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114 If you need your money to double in 8 years, what must the interest rate be if is compounded continually?
Growth & DecayTe
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115 If you need your money to double in 8 years, what must the interest rate be if is compounded annually?
Growth & Decay
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116 If you need your money to double in 8 years, what must the interest rate be if is compounded quarterly?
Growth & Decay
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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
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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
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119 If the half-life of an element is 50 years, at what rate does it decay?
Growth & Decay
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120 If the half-life of an element is 50 years, how much of the element is left in 10 years?
Growth & Decay
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121 If the half-life of an element is 50 years, how much of the element is left in 15 years?
Growth & Decay
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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
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Logistic Functions
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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
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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?
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Trig Functions
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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
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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
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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
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Trigonometric Ratios
sinθ = =opposite sidehypotenuse
opphyp
adjacent sidehypotenuse
adjhypcosθ = =
hypotenuseadjacent side
θ
opposite side
tanθ = opposite sideadjacent side= opp
adj
SOH-CAH-TOA
Trig Functions
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123
3.0 8.5θ
8.0
Trig Functions
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127
7 16θ
14
Trig Functions
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Trigonometric Ratios
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
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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
sinθ = =opposite sidehypotenuse
opphyp
Trig Functions
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Trigonometric Ratios
7.0x
30o
sinθ = =opposite sidehypotenuse
opphyp
sinθ = opphyp
opp = (hyp) (sinθ)
x = (7.0)(sin(30o))
x = (7.0)(0.50)
x = 3.5
Trig Functions
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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;
so we'll use the trig function that relates these three:
9.0
x25o
adjacent sidehypotenuse
adjhypcosθ = =
Trig Functions
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Trigonometric Ratios
9.0
x25o
adj = (hyp)(cosθ)
x = (9.0)(cos(25o))
x = (9.0)(0.91)
x = 8.2
adjacent sidehypotenuse
adjhypcosθ = =
adjhypcosθ =
Trig Functions
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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;
so we'll use the trig function that relates these three:
9.0
x
50o
tanθ = opposite sideadjacent side= opp
adj
Trig Functions
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Trigonometric Ratios
9.0
x
50o
opp = (adj)(tanθ)
x = (9.0)(tan(50o))
x = (9.0)(1.2)
x = 10.8
tanθ = opposite sideadjacent side= opp
adj
tanθ = oppadj
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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
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Converting between Radians and Degrees
Convert degrees to radians Convert radians to degrees
Trig Functions
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136 Convert radians to degrees:
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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
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Graphing cos, sin, & tan
Graph by using values from the table .Since the values are based on a circle, values will repeat.
sin x
x
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Graphing cos, sin, & tan
Graph by using values from the table .Since the values are based on a circle, values will repeat.
tan x
x
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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
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141 Which choice(s) below satisfy the following condition:
Concave Up
A
B
C
D
E
F
G
H
I
J none of the above
Trig Functions
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142 Which choice(s) below satisfy the following condition:
has a relative max
A
B
C
D
E
F
G
H
I
J none of the above
Trig Functions
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143 Which choice(s) below satisfy the following condition:
Decreasing
A
B
C
D
E
F
G
H
I
J none of the above
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Power Functions
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Positive Integer Powers
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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
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Let's first consider n to be a positive integer.
n is odd n is even
Power Functions
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The end behaviors of positive integer powers of a power
function follow the rules from the last unit.
Power Functions
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144 Identify the function(s) that satisfy the following:
A
B
C
D
E
F
GH
I
J
Power Functions
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145 Identify the function(s) that satisfy the following:
AB
C
DE
F
GH
I
J
Power Functions
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146 Identify the function(s) that satisfy the following:
AB
C
DE
F
GH
I
J
Power Functions
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147 Identify the function(s) that satisfy the following:
AB
C
DE
F
G
H
I
J
always decreasing
Power Functions
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Negative Integer Powers
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Now let's consider n to be a negative integer.
Recall what a negative exponent means:
Power Functions
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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
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n is evenn is odd
Power Functions
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Power Functions
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148 Identify the function(s) that satisfy the following:
A
B
C
D
E
F
G
H
I
J
Power Functions
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149 Identify the function(s) that satisfy the following:
A
B
C
D
E
F
GH
I
J
Power Functions
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150 Identify the function(s) that satisfy the following:
A
B
C
D
E
F
GH
I
J
Power Functions
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151 Identify the function(s) that satisfy the following:
A
B
C
D
E
F
GH
I
J
an odd function
Power Functions
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Rational Powers
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is defined as the nth root of x
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Rational Powers
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152 Select the choice that satisfies the following:#### the greatest value at x = 10
A
B
C
D
Rational Powers
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153 Select the choice that satisfies the following? the greatest value at x = -10
AB
C
D
Rational Powers
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154 Select the choice that satisfies the following? the least value at x = 10
A
B
C
D
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155 Select the choice that satisfies the following? the least value at x = -10
A
B
C
D
Rational Powers
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Rational Functions
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In a previous unit, the end behaviors of rational functions was discussed.
3 casesm>n:m<n:
m=n:
Rational Functions
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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
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There are 2 kinds of disconitunity:
Removable Essential
Rational Functions
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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
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Examples: graph the following
Rational Functions
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Examples: graph the following
Rational Functions
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Examples: graph the followingRational Functions
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157 There is a hole at x = c for the equation
Find c.
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