# 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.

Slope-Intercept 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 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

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
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).

**Example 2:**

**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).

**Example 2:**

**Solution:**

## 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.

**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 point-slope 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 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.