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Introduction to Graphing

The Rectangular Coordinate System

Scatterplots and Line Graphs

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.

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

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

If distinct points are plotted in the xy-plane, then the resulting graph is called a scatterplot.

Scatterplots and Line Graphs

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.

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

Line Graphs

Sometimes it is helpful to connect consecutive data points in a scatterplot with line segments.This creates a line graph.

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.

Linear Equations in Two Variables

Basic Concepts

Tables of Solutions

Graphing Linear Equations in Two Variables

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.

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

The ordered pair (3, 4) is NOT a solution.

Tables of Solutions

A table can be used to list solutions to an equation.

A table that lists a few solutions is helpful when graphing an equation.

EXAMPLE Completing a table of solutions

Complete the table for the equation y = 3x – 1.

Solution

x 3 1 0 3

3( 3) 1

x 3 1 0 3

3( ) 1

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.

–1 –3

Plot the points and connect the points with a straight line.

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.

–4 0

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.

EXAMPLE Solve for y and then graphing

Graph the linear equation by solving for y first.

SolutionSolve for y.

–2 1

3 6 12x y

3 6 12x y 6 3 12y x

More Graphing of Lines

Finding Intercepts

Horizontal Lines

Vertical Lines

Finding Intercepts

The y-intercept is where the graph intersects the y-axis.

The x-intercept is where the graph intersects the x-axis.

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 4 12

3( ) 4 12

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.

x 3 1 0 1 3

y 6 4 3 2 0

The x-intercept is (3, 0). The y-intercept is (0, –3).

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(0)

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.

Horizontal Lines

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.

Vertical Lines

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.

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.

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.

Slope and Rates of Change

Finding Slopes of Lines

Slope as a Rate of Change

The rise, or change in y, is y2 y1, and the run, or change in x, is x2 – x1.

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)

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

3 ( 3)

Solution2 1

SlopePositive slope: rises from left to rightNegative slope: falls from left to right

SlopeZero slope: horizontal lineUndefined slope: vertical line

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.

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).

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.

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.

Solutionb. The point (4, 15) means

that after 4 hours the boat is 15 miles from the dock.

Solution

c. The slope is –5. The slope means that the boat is going toward the dock at 5 miles per hour.

Slope-Intercept Form

Finding Slope-Intercept Form

Parallel and Perpendicular Lines

Finding Slope-Intercept Form

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 .

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).

EXAMPLE Graphing an equation in slope-intercept form

Write the y = 4 – 3x equation in slope-intercept form and then graph it.

Solution4 3

Plot the point (0, 4).The line falls 3 units for each 1 unit increase in x.

Parallel and Perpendicular Lines

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)

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

b. The y-intercept is 0. Perpendicular line has a slope of

Point-Slope Form

Derivation of Point-Slope Form

Finding Point-Slope Form

Applications

, 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

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)

The point (4, 3) lies on the line because it satisfies the point-slope form.

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.

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)

If we choose (2, 5), the point-slope form with x1 = 2 and y1 = 5 becomes

15 ( 2).

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.

6 1( 4)y x

6 4y x

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.

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

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.

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

The x-intercept of 5 indicates that the pool is emptied after 5 hours.

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).

1 2 3 4 5 6

Time (hours)

e. The point (2, 6000) indicates that after 2 hours the pool contains 6000 gallons of water.

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Introduction to...

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