Download Review Packet for Math 623 Final Exam

Review Packet for Math 623 Final Exam
Attached is review material for the 2007 Math 623 Final Exam, which covers chapters 1-4
in the UCSMP Functions, Statistics, and Trigonometry textbook. The material for each
chapter includes a chapter overview, a list of “things a student should be able to do,”
sample problems on chapter material, and more detailed notes about the chapter.
Chapter 1 Overview of Making Sense of Data
This chapter presents different statistical measures for data sets with just one variable (not
relations between two or more variables). There are measures of central tendency (mean,
median, mode) and measures of spread (range, interquartile range, standard deviation, and
variance). The chapter also reviews a variety of methods for presenting data, including
histograms, frequency tables, stem-and-leaf plots, and box-and-whisker plots.
You should be able to:
1. Apply the capture-recapture method
2. Interpret and create pie charts, tables, scatter plots, bar graphs, and stemplots
3. Calculate average rates of change
4. Calculate measures of center: mean, median, mode
5. Calculate measures of spread: range, interquartile range, standard deviation, variance
6. Interpret and create 5-number summaries and box-and-whisker plots for datasets.
7. Interpret and create frequency tables and histograms
8. Use sigma notation
1
Problems
Problem 1: A biologist captures and tags 80 frogs from around a network of ponds. He then
releases them and captures a second batch of birds, 90 this time. Of the 90, 12 have tags. What
is an estimate of the frog population?
Problem 2 Find the average rate of change of the population in Boston between 1900 and 1988.
Is the rate of change less than or greater than the rate of change from 1850 to 1970 (Data is in
section 1-2/1-3 below.
A-Block
531
941
9100
88641
Stem
3
4
5
6
7
8
9
F-Block
1
79
29
1334
014
25
Problem 3: Find the minimum, 1st quartile, median, 3rd quartile and maximum scores and the
range for A-block.
Problem 4: The following are the yearly incomes (in $ thousands) of employees at a small
company: 40, 33, 55, 22, 55, 38, 41, 51, 45, 280. Calculate the mean, median, and mode of the
set. Compare the mean and median. If they are significantly different, explain why.
Number
of Players
1
3
3
2
2
1
Height in
Inches
67
69
70
72
74
78
Problem 5: The table to the left provides information
about the height of the players on the varsity basketball
team at a local high school. First, calculate the total
number of players. Use that to calculate the mean,
median, and mode.
Problem 6
A teacher has given each of the 20 students in her math class an ID number from 1 to 20. If x1
represents the score on a test of student 1 and x2 represents the score of student 2, etc., use sigma
notation to represent the mean score of the students in this class.
Problem 7
Do the 5-number summary for this set of data and draw a boxplot:
12, 13, 15, 16, 16, 19, 22, 23, 25, 29, 33, 39, 40, 44
Problem 8
2
Find the standard deviation and variance of this set of data: 5,12, 22, 4, 7. If we replace 22, 4,
and 5 with 10, 10, and 11, what will happen to the mean? What will happen to the standard
deviation?
Chapter 1 Review Material
1.1 Collecting Data
The set of all individuals or objects that you want to study is called the population. The
characteristic that you want to study (height, weight, IQ, etc.) is called a variable. It may not be
possible to study the entire population. In that case, you study only a part of the whole
population, a subset called a sample. When samples are taken randomly from the entire
population that means that every member of the population has an equal chance of being chosen.
Capture-recapture method: this is a method of collecting a sample of objects, animals, or
people from a population. For birds, for example, certain birds, randomly selected, are captured
and tagged. They are then released back into their habitat. A second group of birds is randomly
selected. By counting the number of tagged birds in the second group, you can estimate the
population you studying:
number of tagged animals in sample number of tagged animals in population
=
where P is the
number of animals in sample
P
population you are studying. For example, say you capture and tag 50 birds and then release
them back into the wild. A short period later, you capture another group of 60 birds. If 10 of
those 60 are tagged, you can estimate the population of birds overall is 10/60 = 50/P. Crossmultiply to get 10P = 3000 or P = 300. That is your estimate of the bird population.
1-2 and 1-3: Displays of Data
You should be able to read and interpret tables and a variety of graphs.
Line, Bar, and Scatter Plot Graphs
One type of data set is called time-series data, which shows the change in a variable over time.
Time-series data can be shown in a variety of formats, including:
3
Population (in thousands
)
Population (in thousands
)
Population (in thousands)
900
900
800
700
600
500
400
300
200
100
0
900
800
700
600
500
400
300
200
100
0
1800
1850 1900 1950 1960 1970 1980 1988
800
700
600
500
400
300
200
100
0
1850
A
1900
1950
1850
2000
1900
B
Year
Population (in
thousands)
1950
2000
C
1850
1900
1950
1960
1970
1980
1988
137
561
801
697
641
563
578
Above is a table showing the population of Boston from 1850 to 1988 (time-series data) and
three means of representing the data: A) bar graph B) line graph and C) scatter plot. Both B
and C are coordinate graphs where points and then connected (B) or not connected (A).
These graphs also enable us to calculate average rates of change between any two points on the
graph. For example, the average rate of change of the population of Boston between 1850 and
1970 is calculated by finding the change in the y-variable (population) and dividing by the
641 ! 137
404
=
= 3.37 which, in
change in the x-variable (years) which, in this case, gives us
1970 ! 1850 120
this case, means an average change of 3.37 thousand or 3.370 per year. This is equal to the slope
of the line segment that would connect those two points on the graph.
.
Stem-and-Leaf Plots (or Stemplots)
A stemplot takes a set of data and organizes it by the number in the tens digit. For example,
consider the following set of test scores: 31, 57, 95, 92, 84, 73, 81, 71, 62, 80, 69, 59, 74, 73.
Stems
3
4
5
6
7
8
9
Leaves
1
79
29
1334
014
25
The stemplot puts the tens digit for each score in the
Stems column, arranged from lowest to highest, and
puts each ones digit in the leaves column. Notice that
the score 73 is repeated so it appears twice in the
stemplot. Leaves are arranged from left to right lowest
to highest. If there were a score of 105, it would have
a stem of 10.
With a stemplot, it is easy to find the maximum score (95), the minimum score (31) and the
range (95-31 = 64). Notice as well how easy it is to see that the score of 31 is not clustered with
the other scores and therefore represents an outlier (scores which are very different from the
rest).
4
F-Block
A-Block Stems
531
941
9100
88641
3
4
5
6
7
8
9
1
79
29
1334
014
25
A back-to-back stemplot has the stem in the center
and two sets of leaves, each arranged with the
lowest number nearest the stem and the numbers
rising as you move away from the stem. The
back-to-back plot enables you to compare two sets
of data. To the left is a plot comparing two
classes:
1-4 Measures of Center
Measures of Center or Measures of Central Tendency are meant to represent “typical” scores.
Mean: This is the arithmetic average of the data. Find the sum of the data and divide by the
number of items in the data set. For the F-block class in the stemplot above, the average would
31+57+59+62+69+71+73+73+74+80+81+84+92+95
be
= 71.5
14
Median: If you arrange all the data points in numerical order and take the middle point, the
value of that data point is the median. If there are an odd number of data points, you will have
one middle score which will be the median. If you have an even number of data points, you will
have two middle points. Just take the average of those two points to find the median
For the F-block class above, there are 14 scores, so there are two scores in the middle, 73 and 73.
The median is the average of these two scores, 73.
For the A-block class, there are 15 scores, so there is one middle score, 71, and that is the
median.
Mode: Less commonly used to represent the central tendency of a set of data, the mode is the
value that occurs the most frequently in the set. For example, for the F-block class, the mode is
73 because that is the most commonly occurring score (it is the only that occurs twice). For the
A-block class, there are two modes: 88 and 70, each of which occur twice.
Means and Sigma Notation
If x1, x2, x3, x4, … xn is a set of numbers, then you can represent the sum of that set of numbers
n
as
!x .
i
Since the mean of a set of numbers is the sum divided by the number of items in the
x =1
5
set, you can use the sigma notation to represent the mean as well. The mean of the set referred to
n
!x
i
earlier is
x =1
n
Problem 6
A teacher has given each of the 20 students in her math class an ID number from 1 to 20. If x1
represents the score on a test of student 1 and x2 represents the score of student 2, etc., use sigma
notation to represent the mean score of the students in this class.
1-5 Quartiles, Percentiles, and Box Plots
We will now start looking at measures that represent the spread or distribution of data rather
than the central tendency. Range, which we have seen earlier, measures the difference between
the highest and the lowest values in a set, giving a picture of how spread out the data items are.
Quartiles
Quartiles are the values that divide an ordered set into 4 subsets of equal size. Find the quartiles
as follows:
2nd quartile – the median of the data
1st quartile – the median divides the data set into two parts: numbers below (but not including)
the median and numbers above (but not including) the median. The 1st quartile is the median of
the set of numbers below the median.
3rd quartile – median of the set of numbers above the median of the entire set.
Example: If the ordered (put in order) set is the following: 2,4, 5, 7, 9, 11, 13, 14, 16, 19, 22,
22, 25, 30, 40, what are the 1st, 2nd, and 3rd quartiles?
2nd quartile: 14
2nd quartile is the median. Since there are 15 items in the set, 7 fall into
the lower set and 7 into the upper set and the median is the 8th number or 16 for this set.
1st quartile: 7
1st quartile is the median of the 7 numbers below 14 (the median)
3rd quartile: 22
3rd quartile is the median of the 7 numbers above 14 (the median)
The interquartile range or IQR is the difference between the 1st and 3rd quartiles. In this case,
that would be 22-7 or 15. The IQR tells you the upper and lower values for the middle 50
percent of the data. The 3 quartiles plus the minimum and maximum of the set together give you
the 5-number summary of the data.
Box Plot or Box-and-Whiskers Plot
This is a visual representation of the 5-number summary. For the above set, the plot would look
like this:
6
Box
Minimum
Whiskers
0
10
20
1st Qrt.
Median 3rd Qrt
30
40
50
Maximum
The box refers to the box that starts at the 1st quartile value (8) and extends to the 3rd quartile
value (22). The length of the box is 22-8 = 14 which is the IQR. 50 percent of the scores are
between 8 and 22. The vertical bar all the way to the left represents the minimum score (2) and
the bar all the way to the right represents the maximum score (40). The horizontal lines on either
side of the box represent the “whiskers.” The length of the whole figure, from 2 to 40 is 38 and
that represents the range. The vertical line inside the box is the median.
Problem 7
Do the 5-number summary for this set of data and draw a boxplot:
12, 13, 15, 16, 16, 19, 22, 23, 25, 29, 33, 39, 40, 44
1-6 Histograms
1-8 Variance and Standard Deviation
Standard Deviation and Variance are two additional measures of spread. Both relate to the
mean of a set of data and both are based on the deviation, or difference of each data value from
the mean.
Here is the algorithm for calculating the variance and standard deviation for a data set with n
numbers:
1. Calculate the mean of the data
2. Find the deviation (difference) of each value from the mean
3. Square each deviation and add the squares.
4. Divide the sum of the squared deviations (step 3) by n-1 (not by n). This is the variance.
5. Find the square root of the variance. This is the standard deviation.
Example:
Data set contains elements 3, 4, 7, 10, 12. The mean is (3+4+7+10+12)/5 = 7.2
Number
Deviation from the Mean
Deviation Squared
3
3 – 7.2 = -4.2
-4.22 = 17.64
7
-3.22 = 10.24
-0.22 = 0.04
2.82 = 7.84
4.82 = 23.04
Sum = 60.8
Divide 60.8 by (5-1) = 60.8/4 = 15.2 = Variance of the set
4
7
10
12
4 – 7.2 = -3.2
7 – 7.2 = -0.2
10 – 7.2 = 2.8
12 – 7.2 = 4.8
The standard deviation =!15.2 = 3.899
We can also also write the variance and the standard deviation using sigma notation:
n
for the set x1, x2, ......, xn , we say that the mean = x . The variance = s 2 =
n
" ( x ! x)
" ( x ! x)
2
i
i =1
n !1
2
i
The standard deviation = s =
i =1
n !1
Two sets may have the same mean but substantially different standard deviations. That means
that the set with the higher standard deviation is more spread out.
8
Chapter 2: Overview of Functions and Models
This chapter introduces bivariate data, which mans that it involves relationships between two
variables. Frequently, one of the variables is time as you look at how one variable (population,
average income, etc.) change over time. The chapter looks at several types of mathematical
models – linear and quadratic. These provide mathematical descriptions of real situations,
usually involving some simplification. These models can be used to make estimations and
predictions.
Students should be able to:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Recognize whether relationships are functions; use function notation
Identify domain and range of functions, including absolute value, quadratic, linear, step
Interpret and develop linear models (y=mx + b) to fit real world situations
Create scatter plots, find center of gravity, and line of best fit
Understand limitations of data and interpolation and extrapolation
Work with step functions
Interpret correlation measures (r and r2)
Solve and graph quadratic equations
Find quadratic models to fit data
9
Problem 1
Which of the above graphs represent functions?
Problem 2
f(x) = |x| and g(x) = 12 – x.
a) Evaluate f(-3) + f(3).
b) Evaluate g(-4)*f(– 5)
c) Find the domain and range of f(x) and g(x).
Problem 3
Create a linear model to represent this situation: there is a linear relationship between how long
it takes to react to pain and the distance of the injury from the brain. The reaction time to pain
100 centimeters from the brain is 6.8 milliseconds. The reaction time to pain 170 centimeters
from the brain is 7.5 milliseconds.
Problem 4
Plot the following data points, find the center of gravity and find a line of best fit that goes
through the center of gravity
x
2
5
7
9
10
12
y
5
13
15
20
24
27
Problem 5
For the quadratic model y = 3x2 – 2x – 5
a) Find the y-intercept
b) Find the x-intercepts
c) Find the vertex
d) Find y when x = 4
10
2-1 The Language of Functions.
This section introduces the idea of a function as a particular type of relationship between two sets
of ordered pairs.
Function: Ordered pairs in which each first element is paired with exactly one second element.
A second definition is a correspondence between two sets A and B in which each element of A
corresponds to exactly one element of B.
A function assigns each x value to one y value. If a relation. assigns two different y values to a
value of x, then it is not a function.
You can tell from the graph of a relation if it is a function. For it to be a function, it must pass
the vertical line test. This means that the curve cannot intersect a vertical line in more than one
place (see problem 1)
The set of first elements if the Domain of the function and the set of second elements is the
Range. Generally, you may assume that the domain of a function is the set of all real numbers
for which the function is defined, unless some other domain is explicitly stated.
In many contexts the second number in the ordered pair depends on the first number in the pair.
The first number then is called the independent variable and the second is the dependent
variable.
Function notation: the symbol f(x) is read “f of x” and is also called Euler notation. The
notation is particularly useful when there are multiple functions in a situation and each function
can then be assigned its own name. The number or variable in the parentheses is called the
argument. The function f(x) = 2x – 7 means that you take the argument, multiply it by 2 and
then subtract 7.
Example 1
f(x)=2x –3 and g(x) = x2 + 2
a) Evaluate f(3) + g(2): f(3) means that you replace x with 3 in the expression 2x – 3. This
gives you 2(3) – 3 = 3. g(2) = 22 + 2 = 6. f(3) + g(2) = 3 + 6 = 9.
b) Find the domain and range of f(x) and g(x). For f(x), the domain is the set of real numbers
because the function is defined for all values of x. The range will also be the set of real numbers.
For g(x), the domain is again the set of all real numbers. The range, however, will be restricted
to y>=2 because x2 is always zero or positive and when you add 2, the result will always be 2 or
greater.
11
For the final: be able to identify the domain and range of a function:
There are several issues that come up with domains:
-- The function may not be defined for every value of x. Examples include:
f(x) = 1/x where the function is not defined when x = 0
g(x) = !x where the function is not defined when x<0
h(x) = 1/(x-3) where the function is similar to f(x) but in this case is not defined for x=3.
-- The function may have real world constraints that limit the domain.
j(x) = the total cost if you are buying x CDs at $10 per CD. The domain is restricted to
positive integers because you cannot buy a fraction of a CD or a negative number of CDs.
There are also issues that arise with respect to the range:
-- The range may be limited because the function only produces certain values:
f(x) = floor(x) means that you round down to the next lowest integer. The range is
limited to integers.
g(x) = x2 the range is limited to nonnegative numbers because x2 can not be negative.
h(x) = 1/x – the range excludes 0 because 1/x can never equal 0
2-2 Linear Models
Be able to develop an equation to describe a situation: A carpet installer charges $100 plus $2
per square foot: C = total cost; x = square feet of carpeting: C = 2x + 100
Be able to take the equation from above and solve given a value of x or given a value of C
-- If there are 150 sq feet of carpeting, what is the cost: C = 2(150)+100 = $400
-- If the total cost of the installation is $624, how many sq. feet of carpeting were
installed? 624 = 2x = 100 ! 524 = 2x ! 262 = x
Be able to find an equation given two points or a point and the slope: find equation of line that
passes through (2,5) and (4,12).
-- Slope = (12-5)/(4-2) = 7/2 = 3.5
-- Point Slope form: y-5 = 3.5(x-2) or y-12 = 3.5(x-4). Either is the equation of the line
-- Slope Intercept form: y=mx+b where m is the slope and b is the y-intercept.
Converting y-5=3.5(x – 2) ! y – 5 = 3.5x – 7 ! y = 3.5x – 2.
Be able to explain the meaning of the slope and the y-intercept in a particular word problem: in
the carpenter above, the slope is the charge per square foot of carpet; the y-intercept ($100) is the
setup fee charged by the carpet installer even before any carpet is laid down.
Understand the difference between interpolation and extrapolation: (p. 89)
12
2-3 Line of Best Fit
Be able to calculate center of gravity of a set of coordinates (calculate the mean of the x
coordinates and the mean of the y coordinates). Those two calculations will give you the
coordinates of the center of gravity
Know that a line of best fit must pass through the center of gravity in a linear model.
Be able to use that line of best fit to interpolate and to extrapolate (predict within the range of
date you have and outside the range of your data). Be able to explain the limitations of
extrapolation and provide an example
Know that the sum of the squares of the deviations from the mean of a given set is a measure of
the “spread” of the data in the set. Be able to calculate this (pp. 92-93).
2-4 Step Functions
Know how to work with step functions: how many CDs can you purchase with $54 if each CD
costs $7.50
Know the meanings of continuous, discrete, rounding up, ceiling function
2-5 Correlation
Correlation coefficient measures the strength of the linear relationship between two variables.
Perfect correlation = 1 means that all points fall on the same line.
Positive correlation: dependent variable goes up as independent variable goes up. Both fall at
the same time.
2-6 Quadratic Models
!b ± b 2 ! 4ac
Memorize quadratic formula: if ax + bx + c = 0, then
2a
2
If b2-4ac<0, x has no real value. If b2-4ac=0, then x = -b/2a.
Be able to use Newton’s equation for the height of an object after it has been thrown into the air
(p. 113)
Be able to use a quadratic formula that describes a situation to predict. Be able to interpret r2.
2-7 Finding Quadratic Models
Be able to take 3 points on a line or on a quadratic curve and derive the equation (p. 120-122)
13
Chapter 3 Overview of Transformations of Functions and Data
This chapter focuses on transformations of graphs and of sets of data. Two types of
transformation are explored: translations (adding a constant to one or both variables) and scale
changes (multiplying one or both variables by a constant). The chapter also reviews the
symmetry of functions with respect to the y-axis or to the origin. Finally, Chapter 3 looks at how
to combine functions and to find the inverses of functions.
Students should be able to:
1. Sketch quickly the parent graphs of y = x2, y=x3, y=1/x, y= 1/x2, y=|x|, y = greatest
integer in x (step function), and y = !x
2. Sketch translations of parent graphs, describe the translation using translation notation,
and find the equation of the translated curve
3. Describe the impact on measures of central tendency and measures of spread of adding
a constant to each item in a set of data.
4. Sketch scales changes of parent graphs, describe the scale change using S(x,y) notation,
and find the equation of the transformed curve
5. Describe the impact on measures of central tendency and measures of spread of
multiplying each item in a data set by a constant.
6. Recognize and prove symmetry of functions (showing that a function is odd, even, or
neither).
7. Evaluate compositions of functions
8. Find inverses of functions and identify if two functions are inverses of each other.
14
Problem 1
5
y
y
5
x
-5
-5
5
a
y
5
x
-5
-5
Sketch the graphs of a) y=x2 b) y = 1/x
x
-5
5 -5
b
5
c
c) y = |x|
Problem 2
Find the equations of the images of functions a, b, and c above under the following translations.
Also, sketch each image above on the same graph as the image’s parent function.
For curve a): T(x,y) = (x-1,y+1)
For curve b): T(x.y) = (x – 4, y)
For curve c): T(x,y) = (x + 2, y – 2)
Problem 3
The salaries at Alpha Dog Food Inc., have the following statistical measures: Mean = $45,000;
Median = $50,000; Range = $100,000; Interquartile Range = $20,000; Standard Deviation =
$11,000; Variance = $121,000.
If everyone at the company is given a $1000 bonus at the end of the year, find the effect of that
bonus on each of the statistical measure listed above.
Problem 4 Graph each parent function under the scale change listed below the blank graph.
5
y
5
y
x
-5
-5
5
y
5
x
-5
-5
-5
5 -5
15
x
5
y = x3 S(x,y) = (x/2, 2y)
y = 1/x2
S(x,y) = (2x, -2y)
y = !# x "$
S(x,y) =
(2x,y/2)
Problem 5
Using the same data as in problem 3, assume that instead of a $1,000 bonus, each person in the
company gets a 2% raise. What would be the effect of that increase of each of the statistical
measures given?
Problem 6
Identify each of the following functions as odd, even, or neither:
a) y = x3 – 2x
b) y = (x-2)2 + |x|
c) y = 3x4 - 3
Problem 7
f(x) = 3x2 – 2 g(x) = 3 - x
a) Find f(g(4))
b) Find g°f(0)
c) Show algebraically that f(g(x)) does not equal g(f(x)).
Problem 8
Find the inverses of f(x) and g(x) from problem 7. Are the inverses functions?
16
Chapter 3 Details
Graph Translations and Scale Changes
When a parabola, the graph of y =x2, is shifted 3 units to the left, it create an image parabola
where the x-coordinate of each point is 3 greater than the x-coordinate of the corresponding point
on the original parabola.
The new parabola also has an equation. The point 4,16 is on the original parabola. The image of
that point is 7,16. Whereas the point 4,16 worked with the equation y = x2 (because 16 = 42), the
point 7,16 does not fit that equation. Instead, y is now the square not of x, but of x-3 because y
stayed the same and is still the square of 4 but x is now 7. This will be true of every point on the
original parabola and its image point on the image parabola. Accordingly, the equation of the
new parabola is y = (x-3)2.
If the shift of f(x) is to the right 3 units, the equation of the image function will be y = f(x-3). If
the shift is 3 to the left, the image function equation is y = f(x+3). Similarly, if there is a shift up
of 2, the equation of the image function if y – 2 = f(x), etc.
This works the same with scale changes: If each x-coordinate is doubled, the image equation is
y= f(x/2), etc.
See the attached page for a series of examples of translations and scale changes of graphs.
Translations and Scale Changes of Data
When the same constant is added to each item in a data set, that same constant should be added
to all the measures of central tendency of that dataset: mean, median, and mode. If the mean of
a set is 10 and 5 is added to each member of the set, then the mean of the adjusted set is 15 (10 +
5). However, all measures of spread – range, interquartile range, standard deviation, and
variance, remain unchanged.
When each item in the set is multiplied by the same constant, then you must also multiply the
measures of central tendency by that same constant. For example, if each item in a set with a
mean of 11 and a median of 12 is multiplied by 3, then the mean of the translated data will be 3 x
11 = 33 and the mean of the translated data will be 3 x 12 = 36. Unlike with translations of data,
scale changes of data also change the measures of spread – range, interquartile range, and
standard deviation will all be multiplied by 3 in the example earlier in the paragraph. However,
variance, which is the square of the standard deviation, will be increased not by 3 but by 3x3 or
9.
17
Examples: Translations and Scale Changes on Graphs
Geometric Description of the
Transformation
Transformation
Formula
(using T(x,y) or S(x,y)
x+3
Shifted Left 3
T(x,y) = (x – 3, y)
y=|x|
y + 4 = |x|
Shifted down 4
T(x,y) = (x, y-4)
y = x2
y = (x - 2)2
Shifted Right 2
T(x,y) = (x+2,y)
y = x3
y + 2 = (x+3)2
Shifted Left 3 and Down 2
T(x, y) = (x – 3, y – 2)
Shifted up 1
T(x,y) = (x, y+1)
S(x,y) = (x, 3y)
Parent Function
y=
y=
x
1
x
Transformed
Equation
y=
y=
1
+1
x
y=|x|
y
= |x|
3
Stretch vertically by a factor of
3
y = x2
y = (2x)2
Shrink horizontally by factor of
2
y=
Shrink horizontally by a factor
of 3
y=
y=
x
1
x
y = x3
3x
y
1
=
2 2x
2(y+1)= 3(x –3)3
Stretch vertically by a factor of
2, shrink horizontally by factor
of 2
Shift horizontally right 3 and
vertically down 1; shrunk
vertically by a factor of 2 and
horizontally by factor of 3
&x #
S(x, y) = $ , y !
%2 "
!x "
S(x,y) = # , y $
%3 &
S(x,y) = (x/2, 2y)
Odd and Even Functions
Even Functions: f(-x) = f(x). Graphically, this means that the function is symmetrical around
the y-axis: if reflected across the y-axis, it reflects onto itself.
Examples: Any function with only even exponents or absolute values with no horizontal shifts
(y = x2, y = x4 – x2 + |x|, etc.)
Odd Functions: f(-x) = -f(x). Graphically, this means that the function is symmetrical around
the origin (0,0). It can be rotated 180° around that point and come back onto itself.
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Alternatively, the function can be reflected first across the y-axis and then across the x-axis back
onto itself.
Examples: Any function with only odd exponents with no horizontal shifts: y = x3, y = x5 – x,
etc.
Composition of Functions
When there are two functions f(x) and g(x) and the outputs of one function become the inputs of
the other, that is called a composition of functions. For example, if f(x) = 2x – 4 and g(x) = 3x,
then f(g(5)) means that first g(5) is calculated, = 3 x 5 = 15. Then 15 becomes the input for f(x)
and f(15) = 26. Therefore, f(g(x)) = 26. This composition can also be written f°g(x) which is
read “f following g of x.” The other notation, f(g(x)), Euler’s notation, is read “f of g of x.”
Inverse Functions
Two sets of ordered pairs are inverses of each other if the x coordinates of one set are the ycoordinates of the other set and vice versa. The graphs of the two sets of ordered pairs will be
reflections of each other across the line y=x.
To calculate the inverse of a function, say y = 3x + 2, simply swap the x and y values and solve
for y.
This gives us x = 3y + 2.
Subtract 2 from both sides to get x – 2 = 3y.
x!2
Divide both sides to get y =
. This is the inverse of y = 3x + 2. In this case, both are
3
functions, although that is not always the case.
Two functions, f(x) and g(x), are inverse functions if and only if (f(g(x)) = x for all x in the
domain of g and g(f(x)) = x for all x in the domain of f.
When f and g are inverse functions, then f=g-1
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Chapter 4: Overview of Power, Exponential, and Logarithmic
Functions
In previous chapters, we examined linear and quadratic functions. In this chapter, we will
explore nth root functions (square root, cube root, etc.); rational power functions and properties
of exponents; and exponential functions of the type y=abx, where b>1 means growth and b<1
means decay. We will also introduce the functions that are the inverse of the exponential
functions: logarithmic functions. The chapter also introduces some applications of these
equation types.
You should be able to:
1. Evaluate expressions with fractional exponents and radicals.
2. Recognize and use the Product of Powers Property, Power of a Product Property,
Quotient of Powers Property, and Power of a Quotient Property, Zero Exponent Theorem,
and Negative Exponent Theorem (see section 4-2 below)
3. Graph exponent functions and identify the domain and range.
4. Find an exponential model to fit data.
5. Evaluate logarithms in different bases
6. Use logarithmic equations as inverses of exponential equations and solve for variables in
exponents.
7. Use and evaluate natural logarithms
8. Use natural logarithms in problems; create models using natural logarithms
9. Understand and work with the properties of logarithms (4-7)
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Chapter 4 Problems
1. Evaluate
1
3
a. 625 4
b.
5
32
c. 16 4
d.
3
82
e.
15
2
-3
f. 25 2
2. State the domain and range of y = 10x
3. Find the inverse of each equation:
a. a. y =
5
x3
b. y =
x
4. In 1987, the population of China was approximately 1,062,000,0000. If the population is
growing at a rate of 0.9% per year, what would the population be in 1995? Show the model you
used.
5. If the population of a city is growing at a constant rate and was 750,000 in 1990 and had
reached 1,000,000 by 2000, what is its rate of growth per year?
6. Evaluate:
a. Log2 8
b. Log8 2
c. Log 25
d. Log5 25
e. Log 0.001
7. Carbon-14 has a half-life of 5700 years.
a. Use a continuous growth model to find the annual rate of growth.
b. If an organism had 15 grams of grams of C-14 at death and its fossil contained 1 gram of C14, how old was the fossil?
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4.1 nth Root Functions
nth root of a number x is a solution r to the equation rn=x. When n is an integer>=2, r is an nth
root of x if and only if rn=x. 23 = 8 means that 2 is the 3rd root of 8. Since 34=81 means that 3 is
the 4th root of 81.
The phrase nth root of x can apply to any x, real or complex.
1
The symbol x n is used only when x is a nonnegative real number.
1
n
n
The radical notation x means x when x is nonnegative but also has a use when x is negative
and n is an odd positive integer greater than 1.
1
xn and x n are inverse functions on the domain x: x>=0.
4.2 Rational Power Functions
For all positive integers m and n and base x>0
m
n
1
n m
m
n
1
m n
x = ( x ) = ( n x )m
x = ( x ) = ( n xm )
Example:
7
3
1
3 7
7
1
7 3
1
3
8 = (8 ) = 2 = 128 = (8 ) = 2097152 = 128
Power Postulates
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x m " x n = x m+n
( xy ) n = x n y n
xm
= x m ! n (x # 0)
n
x
n
$x%
xn
& ' = n ( y # 0)
y
( y)
x 0 = 1 ( x ! 0)
1
x!n = n
x
4.3 Exponential Functions
An exponential equation, y=abx, can be defined when a"0, b>0, and b"1
To the left is the graph of y = 3(2x) and
y = 3(0.5x). The two are reflections of
each other across the y-axis.
For both functions, the domain is the
set of all real numbers and the range is
the set of positive real numbers. 3(2x)
is a situation where y increases as x
increases. For 3(0.5x), y decreases as x
increases
Exponential equations describe any situation where growth or decay is at a constant rate.
Frequently, these rates are expressed as percents: the population of the New Zealand grew at an
average annual rate of 3% from 1980 to 1985. This means that, on average, the population of
one year was equal to 1.03 times the population of the previous year. By 1985, the population
had grown from the 1980 population to be 1.03 x 1.03 x 1.03 x 1.03 x 1.03 times as great, or
1.035 times the 1980 population.
When the growth rate, or b, is less than 1, the situation is one of decay. For example, suppose
that you start with 100 grams of a radioactive material that decays at a rate of 50 percent per
year. At the end of x years, you will have 100(0.5)x . If, for example, you want to know how
much is left after 8 years, substitute 8 for x to get 100(0.5)8 which works out to be 0.39 grams.
4-4 Finding Exponential Models
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Suppose that when you graph some given data, it forms a curve that appears to be exponential.
How do you find the equation? As follows:
1. Pick two points. Suppose that you have 74 insects after 3 days and 108 insects after 5 days
and you suspect that the growth is exponential.
2. Create two equations of the y=abx: This gives you 74 = ab3 and 108 = ab5
3. Divide the second equation by the first to get (108/74) on one side and (ab5/ab3). This works
out to 1.4594 = b2 so b = 1.208. Plug this value of b back into the equation to find the value of a
which turns out to be 41.97
4. The equation then is y = 41.97(1.208)x.
By plugging in different values for x, you can predict how many insects there will be after 10
days or 100 days or for any value of x.
4-5 Logarithmic Functions
Although we can find y for different values of x in the exponential equation, we do not know
how to create an inverse equation that would permit us to solve for x given a certain value of y.
We define the inverse of y=bx to be x = by, but how do we solve for y in this second equation?
We define y to be the logarithm of x to the base b if and only if x = by.
Examples:
Log5 125 means “logarithm of 125 base 5” or as an exponential equation, if x = Log5 125, that
means that 5x = 125 so therefore x = 3.
Log8 2 = 1/3 because 8 equals 2 to the 1/3 power.
When a log is written without an explicit base, that means the base is 10. Log 100 = 2 because
102 = 100.
4-6 e and Natural Logarithms
1
The number e is defined as the e = lim (1 + ) n As n gets higher and higher, the expression
n
n!"
approaches the value of e, approximately 2.71828. For a variety of mathematical reasons, it
makes sense to use this number e as the base for logarithms in a number of situations. The
natural logarithm of x is written as ln x.
When a situation is changing continuously (population growth, bacterial division, etc.) it makes
sense to use a model of continuous growth:
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The amount of a substance after t years of growth at the rate r is given by the expression A(t) =
Pert, where P is the initial amount, r is the rate of growth and t is the number of years (or other
time period) that have passed.
Example: If the population of a city has been growing at the rate of 5% annually for 5 years,
from an initial population of 2.5 million, what will be the current population? Since the
population has been growing continuously, the continuous growth model will give us a slightly
more accurate model than the exponential model:
A(5) = 2.5e.05*5 This gives 2.5 * e0.25 = 3.21 million.
Using Logarithms to Solve for Variables
Suppose that we have a situation modeled by the equation y = 3(1.25)x Suppose that we want to
know the value of x when y has a value of 10. We must use logarithms to solve that problem:
1. 10 = 3(1.25)x
2. Divide both side by 3 to get 10/3=(1.25)x
3. Take log of both sides: log(10/3) = log(1.25)x
4. The properties of logarithms enables us to convert the right side of the equation to x(Log
1.25)
5. Divide both sides by Log 1.25 to get ((Log 10/3)/Log 1.25) = x = 5.395. When x = 5.395, y
will equal 10.
Using the Natural Logarithm
Suppose that we know that a city of 500,000 is growing at the rate of 2 percent per year and we
want to know when the population will reach 1,000,000. We would proceed as follows:
1,000,000 = 500,000e.02t where t is the number of years.
-- First, we divide both sides by 500,000 to get 2 = e.02t
-- Second, we take the logarithm of both sides: ln(2) = ln(e.02t) = t lne.02 = .02t
-- Solving, we get 0.693=.02t and therefore t = 34.6 years.
4-7 Properties of Logarithms
These properties of logarithms follows from properties of exponents explored earlier in the
chapter:
•
For any base b, logb 1 = 0
•
For any base b and for any positive real numbers x and y, logb(xy) = logb x + logb y
•
For any base b and for any positive real numbers x and y, logb (x/y) = logb x – logb y
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•
For any base b, any positive real number x and any real number p, logb xp = p logb x
•
For all values of a, b, and c for which logarithms exist, log b a =
26
log c a
log c b