Learn to identify and graph linear equations

Graphing Linear Equations12-1

Learn to identify and graph linear equations.


Vocabulary

linear equation

rate of change


A linear equation is an equation whose solutions fall on a line on the coordinate plane. All solutions of a particular linear equation fall on the line, and all the points on the line are solutions of the equation.


If an equation is linear, a constant change in the x-value corresponds to a constant change in the y-value. The graph shows an example where each time the x-value increases by 3, the y-value increases by 2.


Graph the equation and tell whether it is linear.

y = 3x – 1

Additional Example 1A: Graphing Equations

x 0 1 2 3

y –1 2 5 8

+3 +3 +3

Make a table of ordered pairs. Find the differences between consecutive data points.

+1 +1 +1


Additional Example 1A Continued

The equation y = 3x – 1 is a linear equation because it is the graph of a straight line and each time x increases by 1 unit, y increases by 3 units.


Be careful when graphing each ordered pair. Double check each point you plot.

Caution!



y = x3

Additional Example 1B: Graphing Equations

Make a table of ordered pairs. Find the differences between consecutive data points.

x –2 –1 0 1 2

y –8 –1 0 1 8

+7 +1 +1 +7

+1 +1 +1 +1


Additional Example 1B Continued

The equation y = x3 is not a linear equation because its graph is not a straight line. Also notice that as x increases by a constant of 1 unit, the change in y is not constant.


Additional Example 1C: Graphing EquationsGraph the equation and tell whether it is linear.

y = 2

The equation y = 2 is a linear equation because it is the graph of the horizontal line where every y-coordinate is 2.



y = 2x + 1

Check it Out: Example 1

x –4 –2 0 2 4

y –7 –3 1 5 9

+4 +4 +4 +4

+1 +1 +1 +1Make a table of ordered pairs. Find the differences between consecutive data points.


Check It Out: Example 1 Continued

The equation y = 2x + 1is linear equation because it is the graph of a straight line and each time x increase by 1 unit, y increases by 2 units.


A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable.

The rates for a set of data may vary, or they may be constant.


Determine whether the rates of change are constant or variable.

Additional Example 2A: Identifying Constant and Variable Rates of Change in Data

Find the difference between consecutive data points.

x 0 1 3 5 8

y 0 2 6 10 16

+1 +2 +2 +3

+2 +4 +4 +6

Find each ratio of change in y to change in x.

2 1

= 2 4 2

= 2 4 2

= 2 6 3

= 2

The rates of change are constant.



Additional Example 2B: Identifying Constant and Variable Rates of Change in Data


x 1 3 4 6 9

y 0 2 6 6 3

+2 +1 +2 +3

+2 +4 +0 +3


2 2

= 1 4 1

= 4 0 2

= 0 3 3

= 1

The rates of change are variable.



Check It Out! Example 2A


x 0 1 2 3 4

y 1 3 5 7 11

+1 +1 +1 +1

+2 +2 +2 +4


2 1

= 2 2 1

= 2 2 1

= 2 4 1

= 4

The rates of change are variable.



Check It Out! Example 2B


x 0 1 4 6 9

y 0 2 8 12 18

+1 +3 +2 +3

+2 +6 +4 +6


2 1

= 2 6 3

= 2 4 2

= 2 6 3

= 2

The rates of change are constant.


Additional Example 3: Sports Application

A lift on a ski slope rises according to the equation a = 130t + 6250, where a is the altitude in feet and t is the number of minutes that a skier has been on the lift. Five friends are on the lift. What is the altitude of each person if they have been on the ski lift for the times listed in the table? Draw a graph that represents the relationship between the time on the lift and the altitude.


Additional Example 3 Continued




The altitudes are: Anna, 6770 feet; Tracy, 6640 feet; Kwani, 6510 feet; Tony, 6445 feet; George, 6380 feet. This is a linear equation because when t increases by 1 unit, a increases by 130 units. Note that a skier with 0 time on the lift implies that the bottom of the lift is at an altitude of 6250 feet.



Check It Out: Example 3

In an amusement park ride, a car travels according to the equation D = 1250t where t is time in minutes and D is the distance in feet the car travels. Below is a chart of the time that three people have been in the cars. Graph the relationship between time and distance. How far has each person traveled?

Rider Time

Ryan 1 min

Greg 2 min

Colette 3 min



t D =1250t D (t, D)

1 1250(1) 1250 (1, 1250)

2 1250(2) 2500 (2, 2500)

3 1250(3) 3750 (3, 3750)

The distances are: Ryan, 1250 ft; Greg, 2500 ft; and Collette, 3750 ft.



x

y

This is a linear equation because when t increases by 1 unit, D increases by 1250 units.

1250

2500

1 2

3750

5000

3 4Time (min)

Dis

tan

ce (

ft)


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


Lesson Quiz

Graph each equation and tell whether it is linear.

1. y = 3x – 1

2. y = x

3. y = x2 – 3

yes

yes

no

14


1. Identify the graph of the equation y = 4x – 1 and tell whether it is linear.

A. yes B. yes



2. Identify the graph of the equation and tell whether it is linear.

A. yes B. yes



3. Identify the graph of the equation y = x2 – 9 and tell whether it is linear.

A. no B. no


Documents

Learn to identify and graph linear equations