Graphing and Writing Linear Functions
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.
SlopeIntercept Form
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 slopeintercept form of
a line, where m is the slope and b is the yintercept.
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
use .
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.
Example 2:
Solution:
Method 2: Use the Intercepts
The intercepts are the points where the graph crosses each axis. There are
xintercepts and yintercepts. We find the xintercepts by letting y = 0 and the
yintercepts by letting x = 0.
Example 1: h(x) = 3 x  6
Solution: The yintercept is (0, 6).
To get the xintercept let y = 0.
0 = 3 x  6
6 = 3 x
x = 2
The xintercept is(2, 0).
Example 2:
Solution: The yintercept is (0, 4).
To get the xintercept.
So the xintercept is (6, 0).
Method 3: Use the Slope and yintercept
We can use the slope and intercept of the line to sketch it. Start by plotting
the yintercept and then use the slope
as a map to find a second point.
Example 1: s(x) =4 x + 1
Solution: Start with the yintercept (0, 1), them from move down 4 and
right 1. This gives the point (1,3).
Example 2:
Solution:
Writing Linear Functions
To write the linear function that passes through two points, we'll use the
slopeintercept form f(x) = m x + b. This means we must first find the slope of
the line and then find the yintercept.
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 yintercept 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.
Solution:
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 pointslope form of a line. y  y_{1} = m(x  x_{1})
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.
Solution: .
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
Theorem:
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 slopeintercept 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.
