2.4 Linear Functions
Linear Functions: A linear function is a function that can be written as
The graph of a linear function is a line. To graph a linear function requires at least two points on
the line. Often the easiest points to find are the x and y-intercepts. To find the y-intercept, replace
x with 0 and solve for y; and to find the x intercept replace y with 0 and solve for x.
Graph each linear function. Give the domain and range.
6
1.
y-intercept (x = 0).
0
6
x-intercept (y = 0).
6
The y-intercept is -6 and the point is 0, 6
6
0
6
12
The x-intercept is 12 and the point is 12,0
Domain: ∞, ∞
Range: ∞, ∞
2. 2
5
10
y-intercept (x = 0)
2 0
5
10
5
10
2
y-intercept is 2; point on graph (0,2)
x-intercept (y = 0)
2
5 0
10
2
10
5
x-intercept is 5; point on graph (5,0)
Domain: ∞, ∞
Range: ∞, ∞
The graph of the equation
0 is a line that goes through the origin 0,0 . Since the
origin is both the x and y-intercept, it is necessary to plot an additional point.
3. 3
2
0
0
2
0
3
Domain: ∞, ∞
Range: ∞, ∞
Horizontal and Vertical Lines:
1) The graph of the constant function
or
is a horizontal line that crosses
the y-axis at b.
2) The graph of
is a vertical line that crosses the x-axis at a.
4. 2
4
2
0
4
2 (a vertical line)
Domain: 2
Range: ∞, ∞
5.
3 (a horizontal line)
Domain: ∞, ∞
Range: 3
Every nonvertical line has slope m which is the rate at which the function is either increasing or
decreasing.
Definition: The slope m of a nonvertical line through the points
,
and
Examples: Find the slope of the line going through the given set of points.
6. Through 5, 3
1, 7
7
3
4
1
1 5
4
7. Through
6,7
8. Through 4,9
2, 5
5 7
2
6
12
8
3
2
4,7
The slope is undefined, it’s a vertical line.
9. Find the slope of the line 4
0
3
4
0
0
3
4
0
3
4
3
12 and graph it.
,
is
Examples: Graph the line going through the given point and having the indicated slope.
10. Through
1,3 ,
11. Through
2, 3 ,
12. Through
2,4 ,
0