Graphing and Writing Linear Functions
Definition: A l i n e a r equation can be written in the form Ax + By = C.
Note: These equations are called linear because their graphs are always lines.
Question: Which linear equations are functions?
Answer: All lines except vertical lines are functions.
Definition: A l i n e a r function can be written in the form f(x) = mx + b.
This is also referred to as the slope-intercept form of
a line, where m is the slope and b is the y-intercept.
Note: The slope is the ratio (or comparison) of the change in y divided by the
change in x. A good way to remember slope is to
Graphing Linear Functions
To graph a linear function we need to recall a fact from geometry. We will use
the fact that any two points define a line. Thus to
graph a linear function we need only find two solution points to the linear
equation. There are many ways to accomplish this.
Method 1: Use any Two Points
Example1: f(x) = 2 x - 3
Solution: We need to find two solution points.
Method 2: Use the Intercepts
The intercepts are the points where the graph crosses each axis. There are
x-intercepts and y-intercepts. We find the x-intercepts by letting y = 0 and the
y-intercepts by letting x = 0.
Example 1: h(x) = -3 x - 6
Solution: The y-intercept is (0, -6).
To get the x-intercept let y = 0.
0 = -3 x - 6
6 = -3 x
x = -2
The x-intercept is(-2, 0).
Solution: The y-intercept is (0, 4).
To get the x-intercept.
So the x-intercept is (-6, 0).
Method 3: Use the Slope and y-intercept
We can use the slope and intercept of the line to sketch it. Start by plotting
the y-intercept and then use the slope
as a map to find a second point.
Example 1: s(x) =-4 x + 1
Solution: Start with the y-intercept (0, 1), them from move down 4 and
right 1. This gives the point (1,-3).
Writing Linear Functions
To write the linear function that passes through two points, we'll use the
slope-intercept form f(x) = m x + b. This means we must first find the slope of
the line and then find the y-intercept.
Example 1: Find the equation of the linear function passing through the
two points (2,1) and (0,3). Then write the equation
using function notation.
Solution: First we need to find the slope.
So this tells us that the function is of the form y = -x + b.
To get b, we'll note that the point (0, 3) is the y-intercept and so b = 3.
So our linear function is f(x) = -x + 3
Example 2: Find the equation of the linear function passing through the
two points (3,6) and (-3,2). Then write the equation
using function notation.
So this tells us that the function is of the form
To find b, take one of the points and substitute and then solve.
So the linear function is
Note: There is an alternate way to find the equation of a line. We can
use point-slope form of a line. y - y1 = m(x - x1)
Example 3: Find the equation of the linear function passing through the
two points (5, -3) and (-10, 3). Then write the
equation using function notation.
So the equation is given by .
Now to get this into function form we need to solve for y and simplify.
Parallel and Perpendicular Lines
Two lines are parallel if they have equal slopes.
Two lines are perpendicular if they have opposite reciprocal slopes.
i.e. If the first line has slope , then the
second has slope - .
Example 1: Find the equation of the line passing through the point (1,5)
and parallel to the line 4 x + 2 y = 1.
Solution: First we need to find the slope. Since the new is parallel to 4
x + 2 y = 1, we need to find the slope of this line. We can do this by solving
for y and getting slope-intercept form.
So m = -2.
So our new line is of the form y = -2 x + b.
5 = -2(1) + b
b = 7
So the new linear function is f(x) = -2 x + 7.
Example 2: Find the equation of the line passing through the point (2,1)
and perpendicular to the line 2 x + 3 y = 15.
Solution: We need to find the slope of 2 x + 3 y = 15.
So m =, since the new
line is perpendicular. So it is of the form
So the new function is.