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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
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
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2
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6
8
10
-2
-4
-6
-8
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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
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2
4
6
8
10
-2
-4
-6
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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
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2
4
6
8
10
-2
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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
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2
4
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-2
-4
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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
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-2
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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
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2
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6
8
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-2
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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:
Measuring the Slope of a Line
rise
run
slope = riserun
Linear Functions
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2
4
6
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-2
-4
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-8
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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.
Measuring the Slope of a Line
rise
run
slope = riserun
Linear Functions
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2
4
6
8
10
-2
-4
-6
-8
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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.
Measuring the Slope of a Line
rise
run
slope = riserun
m = 86
m = 43
Linear Functions
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2
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-2
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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.
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
2
4
6
8
10
-2
-4
-6
-8
-10
2 4 6 8 10-2-4-6-8-10 0
Linear Functions
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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.
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)
Linear Functions
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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).
Linear Functions
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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?
Linear FunctionsTe
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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?
Linear Functions
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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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Linear Functions
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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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2 4 6 8 10-2-4-6-8-10 0
Linear Functions
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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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Linear Functions
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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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2 4 6 8 10-2-4-6-8-10 0
Linear Functions
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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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Linear Functions
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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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AB
C
D
Linear Functions
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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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AB
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D
Linear Functions
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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.
Linear Functions
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12 Is 4x -3y = 6 in standard form?
Yes
No
Linear Functions
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13 Is -2x + 3y = 7 in standard form?
Yes
No
Linear Functions
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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
Linear Functions
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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
Linear Functions
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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
Linear Functions
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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
Linear Functions
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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)
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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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Linear Functions
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What are the x- and y- intercepts of y=3x-9?
Linear Functions
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19 Given the equation y = 1/2x-7, what is x when y=0?
Linear Functions
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20 Given the equation y = 1/2x-7, what is y when x=0?
Linear Functions
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21 Given the equation y-4 = 4(x+2), what is x when y=0?
Linear Functions
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22 Given the equation y-4 = 4(x+2) what is y when x=0?
Linear Functions
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Horizontal and Vertical Lines
Return to Table of Contents
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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
Linear Functions
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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
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
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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
Linear Functions
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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
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
Linear Functions
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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
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
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21
(-7,3)
Linear Functions
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35 What is the slope of y-3 = 4(x+2)?
Linear Functions
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36 Which line represents y+5 = -1/3(x-4)?
A line AB line BC line CD line D
2
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AB
CD
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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)
Linear Functions
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Given the equation y = 5(x -1)Determine the point on the line and the slope.
Now graph the line.
Linear Functions
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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.
Linear Functions
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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)
Linear Functions
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39 Which line represents y+6 = -3(x-4)?
A line AB line BC line CD line D 2
4
6
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2 4 6 8 10-2-4-6-8-10 0
A
B
C
D
Linear Functions
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40 Which line represents y+5 = -3(x-4)?
A line AB line BC line CD line D 2
4
6
8
10
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2 4 6 8 10-2-4-6-8-10 0
A B
C
D
Linear Functions
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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)
Linear Functions
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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.
Linear Functions
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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
Linear Functions
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Write the equation of the line with slope of 1/2 and through the point (2,5).
Linear Functions
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Write the equation of the line with slope -3 and containing the point (1,2) in slope-y-intercept form.
Linear Functions
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Write the equation of the line through(5,6) and (7,1).
Linear Functions
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Write the equation of the line through (3,1) and (4,5) in standard form.
Linear Functions
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Write the equation of the line with x-intercept of 5 and y-intercept of 10.
Linear Functions
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Write the equation of the line through (4,1) and parallel to the line y=3x-6.
Linear Functions
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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.
Linear Functions
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42 Which is the equation of the line with slope of -3 and through (1,-4)?
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
Return toTable of
Contents
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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.
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
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
Return toTable of
Contents
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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)
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?
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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?
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:
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