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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Introduction to Graphing
The Rectangular Coordinate System
Scatterplots and Line Graphs
3.1
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Slide 3Copyright © 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.
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Slide 4Copyright © 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 quadrant
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Slide 5Copyright © 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 sales
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If distinct points are plotted in the xy-plane, then the resulting graph is called a scatterplot.
Slide 6Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Scatterplots and Line Graphs
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Slide 7Copyright © 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.
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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 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
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Line Graphs
Slide 9Copyright © 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.
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Slide 10Copyright © 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.
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Slide 11Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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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
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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.
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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.
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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.
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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
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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.
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Slide 18Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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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
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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
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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
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Slide 22Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
More Graphing of Lines
Finding Intercepts
Horizontal Lines
Vertical Lines
3.3
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Slide 24Copyright © 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.
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Slide 25Copyright © 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
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Slide 26Copyright © 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).
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Slide 27Copyright © 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.
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Slide 28Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Horizontal Lines
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Slide 29Copyright © 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.
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Slide 30Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Vertical Lines
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Slide 31Copyright © 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.
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Slide 32Copyright © 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.
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Slide 33Copyright © 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.
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Slide 34Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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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
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Slide 36Copyright © 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.
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Slide 37Copyright © 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
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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
0 (
a.
4 3
7
( )
3)
3
y ym
x x
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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
3 ( 3)
6
b.
( )
1
3
2 4
2
y ym
x x
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Slide 40Copyright © 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
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Slide 41Copyright © 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
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Slide 42Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SlopePositive slope: rises from left to rightNegative slope: falls from left to right
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Slide 43Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SlopeZero slope: horizontal lineUndefined slope: vertical line
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Slide 44Copyright © 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.
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Slide 45Copyright © 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).
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Slide 46Copyright © 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.
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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.
Solutiona. The y-intercept is 35, so
the boat is initially 35 miles from the dock.
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Slide 48Copyright © 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.
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Slide 49Copyright © 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
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Slide 50Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slope-Intercept Form
Finding Slope-Intercept Form
Parallel and Perpendicular Lines
3.5
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Slide 52Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Finding Slope-Intercept Form
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Slide 53Copyright © 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 .
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Slide 54Copyright © 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
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Slide 55Copyright © 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.
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Slide 56Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Parallel and Perpendicular Lines
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Slide 57Copyright © 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
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Slide 58Copyright © 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
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Slide 59Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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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
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Slide 61Copyright © 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
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Slide 62Copyright © 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.
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Slide 63Copyright © 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
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Slide 64Copyright © 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
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Slide 65Copyright © 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
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Slide 66Copyright © 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.
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Slide 67Copyright © 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.
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Slide 68Copyright © 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.
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Slide 69Copyright © 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.