Download Graphing and Analyzing Linear Functions

Graphing and Analyzing Linear Functions
Example 1 Identify Linear Equations
Determine whether each equation is a linear equation. If so, write the equation in standard form.
a. 4 – 2y = 6x
First rewrite the equation so that both variables are on the same side of the equation.
4 – 2y = 6x
Original equation
4 2 y 6x
=
2
2
2 – y = 3x
2 – y + y = 3x + y
2 = 3x + y
Divide each side by the GCF.
Simplify.
Add y to each side.
Simplify.
The equation 2 = 3x + y can be written as 3x + y = 2. The equation is now in standard form where
A = 3, B = 1, and C = 2. This is a linear equation.
b. y + 3 = x2
Since the variable x is raised to the second power, this is not a linear equation.
Example 2
The purchase of a ride card at the zoo allows children to ride the rides and
have their card punched each time. The value of the card is shown in the
graph.
a. Determine the x-intercept, y-intercept, and zero.
The x-intercept is 5 because it is the coordinate where the line crosses the
x-axis. The zero of the function is also 5.
The y-intercept is 20 because it is the coordinate of the point where the line
crosses the y-axis.
b. Describe what the intercepts mean.
The x-intercept 5 means that after 5 rides, the ride ticket has a value of $0.
The y-intercept of 20 means that at time 0, or before any rides were ridden, the ticket had a value of
$20.
Example 3
Tanya is driving 65 miles per hour. The table shows
the function relating the distance remaining on her
trip and the number of hours she has driven.
a. Determine the x-intercept, y-intercept, and zero of
the graph of the function.
x-intercept = 5
5 is the
Time (h) x Distance Remaining (miles) y
0
325
1
260
2
195
3
130
4
65
5
0
value of x when y = 0.
y-intercept = 325
zero = 5
325 is the value of y when x = 0.
The zero of the function is the x-intercept of the graph.
b. Describe what the intercepts mean.
The x-intercept 5 means that after 5 hours, the distance remaining is 0, or she has reached her
destination.
The y-intercept 325 means that she had 325 miles to drive at time 0, or before she started her trip.
Example 4 Graph by Making a Table
Graph y – 3x = -1.
In order to find values for y more easily, solve the equation for y.
y – 3x = -1
Original equation
y – 3x + 3x = -1 + 3x
Add 3x to each side.
y = -1 + 3x
Simplify.
Select values from the domain and make a table. Then graph the ordered pairs. The domain is all real
numbers, so there are infinite solutions. Draw a line through the points.
x
-2
-1
0
1
2
3x – 1
3(-2) – 1
3(-1) – 1
3(0) – 1
3(1) – 1
3(2) – 1
y
-7
-4
-1
2
5
(x, y)
(-2, -7)
(-1, -4)
(0, -1)
(1, 2)
(2, 5)
Example 5
Graph Using Intercepts
Graph 5x – y = 8 using the x-intercept and y-intercept.
To find the x-intercept, let y = 0.
To find the y-intercept, let x = 0.
5x – y = 8
Original equation
5x – y = 8
Original equation
5x – 0 = 8
Replace y with 0.
5(0) – y = 8
Replace x with 0.
5x = 8
Divide each side by 5.
-y = 8
Multiply each side by –1.
x=
8
5
The x-intercept is
y = -8
8
, so the graph intersects
5
(8 )
the x-axis at 5, 0 . The y-intercept is –8, so
the graph intersects the y-axis at (0, -8).
Plot these points. Then draw the line through them.