week 3. due for this week… homework 3 (on mymathlab – via the materials link) monday night at...
TRANSCRIPT
Week 3
Due for this week…
Homework 3 (on MyMathLab – via the Materials Link) Monday night at 6pm.
Read Chapter 5 (The last of the new material for MTH 208)
Do the MyMathLab Self-Check for week 3. Learning team planning for week 5.
Slide 2Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Introduction to Graphing
The Rectangular Coordinate System
Scatterplots and Line Graphs
3.1
Slide 4Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Rectangular Coordinate System
One common way to graph data is to use the rectangular coordinate system, or xy-plane.In the xy-plane the horizontal axis is the x-axis, and the vertical axis is the y-axis.The axes intersect at the origin.The axes divide the xy-plane into four regions called quadrants, which are numbered I, II, III, and IV counterclockwise.
Slide 5Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Plotting points
Plot the following ordered pairs on the same xy-plane. State the quadrant in which each point is located, if possible.
a. (4, 3) b. (3, 4) c. (1, 0)
Solutiona. (4, 3) Move 4 units to the right of the origin and 3 units up.
b. (3, 4) Move 3 units to the left of the origin and 4 units down.
c. (1, 0) Move 1 unit to the left of the origin.
Quadrant I
Quadrant III
Not in any quadrantTry some of Q: 11-20
Slide 6Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Reading a graph
Frozen pizza makers have improved their pizzas to taste more like homemade. Use the graph to estimate frozen pizza sales in 1994 and 2000.
Solutiona. To estimate sales in 1994,
locate 1994 on the x-axis. Then move upward to the data point and approximate its y-coordinate.
b. To estimate sales in 2000, locate 2000 on the x-axis. Then move upward to the data point and approximate its y-coordinate.
a. about $2.1 billion in sales
b. about $3.0 billion in salesTry some of Q: 39-40
If distinct points are plotted in the xy-plane, then the resulting graph is called a scatterplot.
Slide 7Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Scatterplots and Line Graphs
Slide 8Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Making a scatterplot of gasoline prices
The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These price have not been adjusted for inflation.
Year 1975 1980 1985 1990 1995 2000 2005
Cost (per gal in cents)
56.7 119.1 111.5 114.9 120.5 156.3 186.6
The data point (1975, 56.7) can be used to indicate the average cost of a gallon of gasoline in 1975 was 56.7 cents. Plot the data points in the xy-plane.
Slide 9Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Making a scatterplot of gasoline prices
The table lists the average price of a gallon of gasoline for selected years. Make a scatterplot of the data. These prices have not been adjusted for inflation.
Year 1975 1980 1985 1990 1995 2000 2005
Cost (per gal in cents)
56.7 119.1 111.5 114.9 120.5 156.3 186.6
Try some of Q: 23-32
Line Graphs
Slide 10Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Sometimes it is helpful to connect consecutive data points in a scatterplot with line segments.This creates a line graph.
Slide 11Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Making a line graph
Use the data in the table to make a line graph.
x 3 2 1 0 1 2 3
y 3 4 0 3 2 4 3
Plot the points and then connect consecutive points with line segments.
Try some of Q: 33-38
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Linear Equations in Two Variables
Basic Concepts
Tables of Solutions
Graphing Linear Equations in Two Variables
3.2
Slide 13Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Basic Concepts
Equations can have any number of variables.
A solution to an equation with one variable is one number that makes the statement true.
Slide 14Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Testing solutions to equations
Determine whether the given ordered pair is a solution to the given equation.
a. y = x + 5, (2, 7) b. 2x + 3y = 18, (3, 4)
Solutiona. y = x + 5 b. 2x + 3y = 18
7 = 2 + 5
7 = 7 True
The ordered pair (2, 7) is a solution.
2(3) + 3(4) = 18
6 12 = 18
6 18
The ordered pair (3, 4) is NOT a solution.Try some of Q:9-18
Tables of Solutions
Slide 15Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
A table can be used to list solutions to an equation.
A table that lists a few solutions is helpful when graphing an equation.
Slide 16Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Completing a table of solutions
Complete the table for the equation y = 3x – 1.
Solution
x 3 1 0 3
y
3x
3 1
3( 3) 1
9 1
10
y x
y
y
y
x 3 1 0 3
y 10
1x
3 1
3( ) 1
3
4
1
1
y x
y
y
y
0x
3 1
3( ) 1
0
1
0
1
y x
y
y
y
3x
3 1
3( ) 1
9
8
3
1
y x
y
y
y
4 1 8
Try some of Q:19-24
Slide 17Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Graphing an equation with two variables
Make a table of values for the equation y = 3x, and then use the table to graph this equation.
SolutionStart by selecting a few convenient values for x such as –1, 0, 1, and 2. Then complete the table.
x y
–1 –3
0 0
1 3
2 6
Plot the points and connect the points with a straight line. Try some of Q: 35-40
Slide 18Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 19Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Graphing linear equations
Graph the linear equation.
SolutionBecause this equation can be written in standard form, it is a linear equation. Choose any three values for x.
x y
–4 0
0 1
4 2
Plot the points and connect the points with a straight line.
11
4y x
Slide 20Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Graphing linear equations
Graph the linear equation.
SolutionBecause this equation can be written in standard form, it is a linear equation. Choose any three values for x.
x y
0 5
2 3
5 0
Plot the points and connect the points with a straight line.
5x y
Try some of Q: 41-56
Slide 21Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Solve for y and then graphing
Graph the linear equation by solving for y first.
SolutionSolve for y.
x y
–2 1
0 2
2 3
Plot the points and connect the points with a straight line.
3 6 12x y
3 6 12x y 6 3 12y x
12
2y x
Try some of Q: 57-68
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
More Graphing of Lines
Finding Intercepts
Horizontal Lines
Vertical Lines
3.3
Slide 23Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Finding Intercepts
The y-intercept is where the graph intersects the y-axis.
The x-intercept is where the graph intersects the x-axis.
Slide 24Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using intercepts to graph a line
Use intercepts to graph 3x – 4y = 12.
SolutionThe x-intercept is found by letting y = 0.
The graph passes through the two points (4, 0) and (0, –3).
The y-intercept is found by letting x = 0.
3 4 12
3 4( ) 12
3 12
( , )
0
0
4
4
x y
x
x
x
3 4 12
3( ) 4 12
4 12
(0, 3
3
0
)
x y
y
y
x
Try some of Q: 25-44
Slide 25Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using a table to find intercepts
Complete the table. Then determine the x-intercept and y-intercept for the graph of the equation x – y = 3.SolutionFind corresponding values of y for the given values of x.
3
3
3
6
6
x y
y
y
y
x 3 1 0 1 3
y
1
3
3
4
4
x y
y
y
y
3
3
3
3
0
x y
y
y
y
3
3
2
2
1
x y
y
y
y
3
3
0
3
0
x y
y
y
y
x 3 1 0 1 3
y 6 4 3 2 0
The x-intercept is (3, 0). The y-intercept is (0, –3).
Try some of Q: 21-24
Slide 26Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Modeling the velocity of a toy rocket
A toy rocket is shot vertically into the air. Its velocity v in feet per second after t seconds is given by v = 320 – 32t. Assume that t ≥ 0 and t ≤ 10.a. Graph the equation by finding the intercepts.b. Interpret each intercept.
Solutiona. Find the intercepts.
320 32
320 32
320
0
2
0
3
1
v t
t
t
t
320 32
320 32(0)
320
v t
v
v
b. The rocket had velocity of 0 feet per second after 10 seconds. The v-intercept indicates that the rocket’s initial velocity was 320 feet per second.
Try some of Q: 85-86
Slide 27Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Horizontal Lines
Slide 28Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Graphing a horizontal line
Graph the equation y = 2 and identify its y-intercept.
Solution
The graph of y = 2 is a horizontal line passing through the point (0, 2), as shown below.The y-intercept is 2.
Try some of Q: 47-54a’s
Slide 29Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Vertical Lines
Slide 30Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Graphing a vertical line
Graph the equation x = 2, and identify its x-intercept.
Solution
The graph of x = 2 is a vertical line passing through the point (2, 0), as shown below.The x-intercept is 2.
Try some of Q: 47-54 b’s
Slide 31Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Writing equations of horizontal and vertical lines
Write the equation of the line shown in each graph.a. b.
Solutiona. The graph is a horizontal line.
The equation is y = –1.
b. The graph is a vertical line.The equation is x = –1.
Try some of Q: 55-62
Slide 32Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Writing equations of horizontal and vertical lines
Find an equation for a line satisfying the given conditions.a. Vertical, passing through (3, 4).b. Horizontal, passing through (1, 2).c. Perpendicular to x = 2, passing through (1, 2).
Solutiona. The x-intercept is 3. The equation is x = 3.
b. The y-intercept is 2.The equation is y = 2.
c. A line perpendicular to x = 2 is a horizontal line with y-intercept –2. The equation is y = 2.
Try some of Q: 73-80
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slope and Rates of Change
Finding Slopes of Lines
Slope as a Rate of Change
3.4
Slide 34Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slope
The rise, or change in y, is y2 y1, and the run, or change in x, is x2 – x1.
Slide 35Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Calculating the slope of a line
Use the two points to find the slope of the line. Interpret the slope in terms of rise and run.Solution
The rise is 3 units and the run is –4 units.
(–4, 1)
(0, –2)
2 1
2 1
( )
4 0
4
3
2
3
4
1
y ym
x x
Try some of Q: 20-26
Slide 36Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Calculating the slope of a line
Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)
Solution2 1
2 1
0 (
a.
4 3
7
( )
3)
3
y ym
x x
Slide 37Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Calculating the slope of a line
Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)
Solution2 1
2 1
3 ( 3)
6
b.
( )
1
3
2 4
2
y ym
x x
Slide 38Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Calculating the slope of a line
Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)
Solution2 1
2 1
c.
( )
0
)
4
4
0
2 2
4
(
y ym
x x
Slide 39Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Calculating the slope of a line
Calculate the slope of the line passing through each pair of points. a. (3, 3), (0, 4) b. (3, 4), (3, 2)c. (2, 4), (2, 4) d. (4, 5), (4, 2)
Solution2 1
2 1
2 5
d.
( )
undef
4 4
0ed
3in
y ym
x x
Try some of Q: 35-50
Slide 40Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SlopePositive slope: rises from left to rightNegative slope: falls from left to right
Slide 41Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SlopeZero slope: horizontal lineUndefined slope: vertical line
Slide 42Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Finding slope from a graph
Find the slope of each line. a. b.
Solution
a. The graph rises 2 units for each unit of run m = 2/1 = 2.
b. The line is vertical, so the slope is undefined.
Try some of Q: 15-19
Slide 43Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Sketching a line with a given slope
Sketch a line passing through the point (1, 2) and having slope ¾.
SolutionStart by plotting (1, 2).
The slope is ¾ which means a rise (increase) of 3 and a run (horizontal) of 4.
The line passes through the point (1 + 4, 2 + 3) = (5, 5).
Try some of Q: 57-58
Slide 44Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slope as a Rate of Change
When lines are used to model physical quantities in applications, their slopes provide important information.
Slope measures the rate of change in a quantity.
Slide 45Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Interpreting slope
The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.a. Find the y-intercept. What does the y-intercept represent?b. The graph passes through the point (4, 15). Discuss the meaning of this point.c. Find the slope of the line. Interpret the slope as a rate of change.
Solutiona. The y-intercept is 35, so
the boat is initially 35 miles from the dock.
Slide 46Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Interpreting slope
The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.a. Find the y-intercept. What does the y-intercept represent?b. The graph passes through the point (4, 15). Discuss the meaning of this point.c. Find the slope of the line. Interpret the slope as a rate of change.
Solutionb. The point (4, 15) means
that after 4 hours the boat is 15 miles from the dock.
Slide 47Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Interpreting slope
The distance y in miles that a boat is from the dock on a fishing expedition after x hours is shown below.a. Find the y-intercept. What does the y-intercept represent?b. The graph passes through the point (4, 15). Discuss the meaning of this point.c. Find the slope of the line. Interpret the slope as a rate of change.
Solution
c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.
15 05
4 7m
Try some of Q: 93-94
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slope-Intercept Form
Finding Slope-Intercept Form
Parallel and Perpendicular Lines
3.5
Slide 49Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Finding Slope-Intercept Form
Slide 50Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Using a graph to write the slope-intercept form
For the graph write the slope-intercept form of the line.
SolutionThe graph intersects the y-axis at 0, so the y-intercept is 0.The graph falls 3 units for each 1 unit increase in x, the slope is –3.The slope intercept-form of the line is y = –3x .
Try some of Q: 17-26
Slide 51Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Sketching a line
Sketch a line with slope ¾ and y-intercept −2. Write its slope-intercept form.
SolutionThe y-intercept is (0, −2). Slope ¾ indicates that the graph rises 3 units for each 4 units run in x. The line passes through the point (4, 1).
32
4y x
Try some of Q: 27-36
Slide 52Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Graphing an equation in slope-intercept form
Write the y = 4 – 3x equation in slope-intercept form and then graph it.
Solution4 3
3 4
y x
y x
Plot the point (0, 4).The line falls 3 units for each 1 unit increase in x.
Try some of Q: 51-60
Slide 53Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Parallel and Perpendicular Lines
Slide 54Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Finding parallel lines
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2, 1). Sketch a graph of each line.
SolutionThe line has a slope of 3 any parallel line also has slope 3.Slope-intercept form: y = 3x + b. The value of b can be found by substituting the point (2, 1) into the equation. 3
1 3(2)
1 6
5
y x b
b
b
b Try some of Q: 67-74
Slide 55Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Finding perpendicular lines
Find the slope-intercept form of a line passing through the origin that is perpendicular to each line. a. y = 4x b.
Solutiona. The y-intercept is 0. Perpendicular line has a slope of
1.
4
23
5 y x
1
4y x
b. The y-intercept is 0. Perpendicular line has a slope of
5.
25
2y x
Try some of Q: 71-74
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Point-Slope Form
Derivation of Point-Slope Form
Finding Point-Slope Form
Applications
3.6
Slide 57Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
, 0 and 1,xf x a a a
The line with slope m passing through the point (x1, y1) is given by
y – y1 = m(x – x1),
or equivalently, y = m(x – x1) + y1
the point-slope form of a line.
POINT-SLOPE FORM
Slide 58Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Finding a point-slope form
Find the point-slope form of a line passing through the point (3, 1) with slope 2. Does the point (4, 3) lie on this line?
Let m = 2 and (x1, y1) = (3,1) in the point-slope form.
To determine whether the point (4, 3) lies on the line, substitute 4 for x and 3 for y.
y – y1 = m(x – x1)
y − 1 = 2(x – 3)
3 – 1 ? 2(4 – 3)
2 = 2
The point (4, 3) lies on the line because it satisfies the point-slope form. Try some of Q: 9-24
Slide 59Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Finding an equation of a line
Use the point-slope form to find an equation of the line passing through the points (−2, 3) and (2, 5).
Before we can apply the point-slope form, we must find the slope.
2 1
2 1
y ym
x x
5 3
2 2
2
4
1
2
Slide 60Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE continued
We can use either (−2, 3) or (2, 5) for (x1, y1) in the point-slope form. If we choose (−2, 3), the point-slope form becomes the following.
y – y1= m(x – x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes
15 ( 2).
2y x
Try some of Q: 25-30
Slide 61Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Finding equations of lines
Find the slope-intercept form of the line perpendicular to passing through the point (4, 6).
The line has slope m1 = 1. The slope of the perpendicular line is m2 = −1. The slope-intercept form of a line having slope −1 and passing through (4, 6) can be found as follows.
3,y x
6 1( 4)y x
3y x
6 4y x
10y x Try some of Q: 45-54
Slide 62Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Modeling water in a pool
A swimming pool is being emptied by a pump that removes water at a constant rate. After 1 hour the pool contains 8000 gallons and after 4 hours it contains 2000 gallons.
a. How fast is the pump removing water?b. Find the slope-intercept form of a line that models
the amount of water in the pool. Interpret the slope.c. Find the y-intercept and the x-intercept. Interpret
each.d. Sketch the graph of the amount of water in the
pool during the first 5 hours.e. The point (2, 6000) lies on the graph. Explain its
meaning.
Slide 63Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE continued
a. The pump removes 8000 − 2000 gallons of water in 3 hours, or 2000 gallons per hour.
b. The line passes through the points (1,8000) and (4, 2000), so the slope is
Solution
2000 80002000
4 1
Use the point-slope form to find the slope-intercept form.
y – y1= m(x – x1)
y – 8000 = −2000(x – 1)
y – 8000 = −2000x + 2000
y = −2000x + 10,000
A slope of −2000, means that the pump is removing 2000 gallons per hour.
Slide 64Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE continued
c. The y-intercept is 10,000 and indicates that the pool initially contained 10,000 gallons. To find the x-intercept let y = 0 in the slope-intercept form.
0 2000 10,000x 2000 10,000x 2000 10,000
2000 2000
x
5x
The x-intercept of 5 indicates that the pool is emptied after 5 hours.
Slide 65Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE continued
d. The x-intercept is 5 and the y-intercept is 10,000. Sketch a line passing through (5, 0) and (0, 10,000).
X
Y
1 2 3 4 5 6
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0
Wat
er (
gallo
ns)
Time (hours)
e. The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water.
Try some of Q: 63-64
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Introduction to Modeling
Basic Concepts
Modeling Linear Data
3.7
Slide 67Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Generally, mathematical models are not exact representations of data.
Slide 68Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Determining whether a model is exact
A person can vote in the United States at age 18 or over. The table shows the voting-age population P in millions for selected years x. Does the equation P = 3.25x – 6297 model the data exactly? Explain.
To determine whether the equation models the data exactly, let x = 2000, 2002, and 2004 in the given equation.
x 2000 2002 2004
P 203 211 216Source: U.S. Census Bureau
x = 2000: P = 3.25(2000) – 6297 = 203
x = 2002: P = 3.25(2002) – 6297 = 209.5
x = 2004: P = 3.25(2004) – 6297 = 216
The model is not exact because it does not predict the voting-age population of 211 million in 2002.
Try some of Q: 15-20
Slide 69Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Determining gas mileage
The table shows the number of miles y traveled by a motorhome on x gallons of gasoline.
a. Plot the data in the xy-plane. Be sure to label each axis.
b. Sketch a line that models the data.c. Find the equation of the line and interpret the slope
of the line.d. How far could this motorhome travel on 15 gallons
of gasoline?
x 3 6 9
P 24 48 72
Slide 70Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Determining gas mileage
x 3 6 9
P 24 48 72
a. Plot the points (3, 24), (6, 48), and (9, 72).
X
Y
1 2 3 4 5 6 7 8 9 10
10
20
30
4050
60
70
80
90
100
0
Dis
tanc
e (m
iles)
Gasoline (gallons)
b. Sketch a line through the points.
X
Y
1 2 3 4 5 6 7 8 9 10
10
20
30
4050
60
70
80
90
100
0
Dis
tanc
e (m
iles)
Gasoline (gallons)
X
Y
1 2 3 4 5 6 7 8 9 10
10
20
30
4050
60
70
80
90
100
0
Dis
tanc
e (m
iles)
Gasoline (gallons)
X
Y
1 2 3 4 5 6 7 8 9 10
10
20
30
4050
60
70
80
90
100
0
Dis
tanc
e (m
iles)
Gasoline (gallons)
Slide 71Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
continued
c. Find the slope of the line. 72 24
9 3m
48
6 8
Now find the equation of the line passing through (3, 24) with the slope of 8.
y – y1 = m(x – x1)
y – 24 = 8(x – 3)
y – 24 = 8x – 24
y = 8x
The data are modeled by the equation y = 8x. Slope 8 indicates that the mileage of the motorhome is 8 miles per gallon.
d. On 15 gallons of gasoline, the motorhome could go y = 8(15) = 120 miles.Try Q: 53
Slide 72Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Modeling linear data
The table contains ordered pairs that can be modeled approximately by a line.
a. Plot the data. Could a line pass through all five points?
b. Sketch a line that models the data and determine its equation.
x 1 2 3 4 5
y 5 7 9 10 11
Solution
a.
b. One possibility of the line is shown.
X
Y
1 2 3 4 5 6
1
2
3
4
5
6
7
8
9
10
11
12
0
11 5
5 1m
6
4
3
2
y – y1 = m(x – x1)
y – 5 = 3/2(x – 1)3 7
2 2y x
Try some of Q: 33-40
Slide 73Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Modeling with linear equations
Find a linear equation in the form y = mx + b that models the quantity y after x days.
a. A quantity y is initially 750 and increases at a rate of 4 per day.
b. A quantity y is initially 2300 and decreases at a rate of 50 per day.
c. A quantity y is initially 17,875 and remains constant.
SolutionIn the equation y = mx + b, the y-intercept b represents the initial amount and the slope m represents the rate of change.
b. y = −50x + 2300a. y = 4x + 750 c. y = 17,875
Try some of Q: 27-32
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Functions and Their Representations
Basic Concepts
Representations of a Function
Definition of a Function
Identifying a Function
Graphing Calculators (Optional)
8.1
Slide 75Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
, 0 and 1,xf x a a a
The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.
Name
y = f(x)
Output Input
FUNCTION NOTATION
Slide 76Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.
Representations of a Function
Verbal Representation (Words)
Numerical Representation (Table of Values)
Symbolic Representation (Formula)
Graphical Representation (Graph)
Diagrammatic Representation (Diagram)
Slide 77Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 78Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Calculating sales tax
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3).
a. Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars.
b. Numerical Representation x f(x)
$1.00 $0.06
$2.00 $0.12
$3.00 $0.18
$4.00 $0.24
Slide 79Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE continued
c. Symbolic Representation f(x) = 0.06x
d. Graphical Representation
X
Y
1 2 3 4 5 6
0.1
0.2
0.3
0.4
0.5
0.6
0
e. Diagrammatic Representation
1 ●2 ●3 ●4 ●
● 0.06● 0.12● 0.18● 0.24
f(3) = 0.18
Try some of Q: 51-60
Slide 80Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Evaluating symbolic representation (formulas)
Evaluate the function f at the given value of x.
f(x) = 5x – 3 x = −4
f(−4) = 5(−4) – 3
= −20 – 3
= −23
Try some of Q: 19-30
Slide 81Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair:
(x, y).
Input Output
A relation is a set of ordered pairs, and a function is a special type of relation.
Slide 82Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value.
Function
The domain of f is the set of all x-values, and the range of f is the set of all y-values.
Slide 83Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Finding the domain and range graphically
Use the graph of f to find the function’s domain and range.
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
The arrows at the ends of the graph indicate that the graph extends indefinitely. Thus the domain includes all real numbers. The smallest y-value on the graph is y = −4. Thus the range is y ≥ −4.
Slide 84
Try some of Q: 79-86
Use f(x) to find the domain of f.
a. f(x) = 3x b.
Slide 85Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Finding the domain of a function
1
4f x
x
Solution
Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers.
a.
b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4. Try some of Q: 89-102
Determine whether the table of values represents a function.
Slide 86Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Determining whether a table represents a function
x f(x)
2 −6
3 4
4 2
3 −1
1 0
Solution
The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.
Try some of Q: 123-124
Slide 87Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If every vertical line intersects a graph at no more than one point, then the graph represents a function.
Vertical Line Test
Determine whether the graph represents a function.
Slide 88Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Determining whether a graph represents a function
Solution
Any vertical line will cross the graph at most once. Therefore the graph does represent a function.
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
Try some of Q: 113-118
Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Linear Functions
Basic Concepts
Representations of Linear Functions
Modeling Data with Linear Functions
The Midpoint Formula (Optional)
8.2
Slide 90Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
, 0 and 1,xf x a a a
A function f defined by f(x) = ax + b, where a and b are constants, is a linear function.
LINEAR FUNCTION
Slide 91Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Identifying linear functions
Determine whether f is a linear function. If f is a linear function, find values for a and b so that f(x) = ax + b.
a. f(x) = 6 – 2x b. f(x) = 3x2 – 5
Let a = –2 and b = 6. Then f(x) = −2x + 6, and f is a linear function.
a.
b. Function f is not linear because its formula contains x2. The formula for a linear function cannot contain an x with an exponent other than 1.
Try some of Q: 9-16
Slide 92Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Determining linear functions
Use the table of values to determine whether f(x) could represent a linear function. If f could be linear, write the formula for f in the form f(x) = ax + b.
For each unit increase in x, f(x) increases by 7 units so f(x) could be linear with a = 7. Because f(0) = 4, b = 4. thus f(x) = 7x + 4.
x 0 1 2 3
f(x) 4 11 18 25
Try some of Q: 21-28
Slide 93Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Graphing a linear function by hand
Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).
Begin by creating a table.
x y
−1 −4
0 −3
1 −2
2 −1
Plot the points and sketch a line through the points.
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
Slide 94Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
continued
Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).
To evaluate f(4), first find x = 4 on the x-axis. Then find the corresponding y-value. Thus f(4) = 1.
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
0
Try some of Q: 49-58
Slide 95Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
, 0 and 1,xf x a a a
The formula f(x) = ax + b may be interpreted as follows.
f(x) = ax + b(New amount) = (Change) + (Fixed amount)
When x represents time, change equals (rate of change) × (time).
f(x) = a × x + b(Future amount) = (Rate of change) × (Time) + (Initial amount)
MODELING DATA WITH A LINEAR FUNCTION
Slide 96Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE
Solution
Modeling the cost of a truck rental
Suppose that a moving truck costs $0.25 per mile and a fixed rental fee of $20. Find a formula for a linear function that models the rental fees.
Total cost is found by multiplying $0.25 (rate per mile) by the number of miles driven x and then adding the fixed rental fee (fixed amount) of $20. Thus f(x) = 0.25x + 20.
Slide 97Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE Modeling with a constant function
The temperature of a hot tub is recorded at regular intervals.
Discuss the temperature of the water during this time interval.
a.
b. Find a formula for a function f that models these data.
c. Sketch a graph of f together with the data.
Elapsed Time (hours) 0 1 2 3
Temperature 102°F 102°F 102°F 102°F
Slide 98Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE continued
The temperature appears to be a constant 102°F. a.
b. Because the temperature is constant, the rate of change is 0. Thus f(x) = 0x + 102 or f(x) = 102.
c. Graphing the data points, gives the following constant function.
Elapsed Time (hours) 0 1 2 3
Temperature 102°F 102°F 102°F 102°F
Solution
0
20
40
60
80
100
120
0 1 2 3
Time (hours)
Te
mp
era
ture
(d
eg
ree
s)
Try some of Q: 111-112
End of week 3
You again have the answers to those problems not assigned
Practice is SOOO important in this course. Work as much as you can with MyMathLab, the
materials in the text, and on my Webpage. Do everything you can scrape time up for, first the
hardest topics then the easiest. You are building a skill like typing, skiing, playing a
game, solving puzzles. NEXT TIME: Exponents and Polynomials