
## Subsection1.7Linear Functions

Linear relationships are relationships in which the rate of change is constant.

###### Linear Equation

A linear function is a function which has a constant rate of change.

Many phenomena can be modeled using linear functions $y = f (x)$ where the equations have the form

\begin{equation*} f (x) = (\text{starting value}) + (\text{rate of change}) \cdot x. \end{equation*}

The initial value, or the value of $f(0)\text{,}$ is the vertical intercept of the graph, and the rate of change is the slope of the graph. Thus, we can write an equation of a line as

\begin{equation*} f (x) = b + mx \end{equation*}

where the constant term, $(0,b)\text{,}$ is the vertical intercept of the line, and $m\text{,}$ the coefficient of $x\text{,}$ is the slope of the line. This form for an equation of a line is called the slope-intercept form.

###### Slope-Intercept Form

If we write an equation of a linear function in the form,

\begin{equation*} f (x) = mx + b \end{equation*}

then $m$ is the slope of the line, and $(0,b)$ is the vertical intercept although we frequently refer to the intercept by the value $b$ since the other coordinate is always $0\text{.}$

###### Example1.14

An icicle grows according to the formula $H(t)=0.05t+0.12\text{,}$ where $t$ is the time in minutes since the first measurement was take, and $H(t)$ is the height of the icicle in centimeters.

1. The slope is 0.05, which tells us that the icicle's height grows by 0.05 cm each minute.

2. The $y$-intercept is 0.12, which tells us that the height of the icicle was 0.12 cm at the first measurement.

###### Example1.15

Samantha owns a catering business. For any party with up to 100 guests, she charges $\ 2,000\text{.}$ She charges $\ 8$ per person for each additional guest over 100. Give a formula for the cost of having Samantha cater your party as a function of the number of guests over 200.

Solution

A possible formula for the cost of having Samantha cater your party is $C(g)=2000+8g\text{,}$ where $g$ is the number of guests over 100.

The following formula provides a method for finding the value of the slope $m$ when given two points on the line.

###### Two-Point Slope Formula

The slope of the line passing through the points $P_1 (x_1, y_1)$ and $P_2 (x_2, y_2)$ is given by

\begin{equation*} m = \frac{\Delta y}{\Delta x}= \frac{y_2 - y_1}{x_2 - x_1} \text{, }~x_2 \ne x_1. \end{equation*}
###### Example1.16

Compute the slope of the line between the points $(6, -3)$ and $(4, 3)\text{.}$

Solution

Substitute the coordinates of $Q_1$ and $Q_2$ into the slope formula to find

\begin{equation*} m = \frac{y_2 - y_1}{x_2 - x_1}= \frac{3 - (-3)}{4 - 6} = \frac{6}{-2}= -3. \end{equation*}

This value for the slope, $-3\text{,}$ is the same value found above.

Sometimes you will be given the slope of a line and another point on that line. The following formula is helpful in that situation.

###### Point-Slope Form

An equation of the line that passes through the point $(x_1, y_1)$ and has slope $m$ is

\begin{equation*} y- y_1= + m(x- x_1). \end{equation*}
###### Example1.17

Find a formula for the line that has slope -2 and passes through the point $(-1,4)\text{.}$

Solution

Using point-slope form, we have the line $y=4+(-2)(x-(-1))\text{.}$

We can also write this in slope-intercept form by simplifying: $y=2-2x\text{.}$

Is is also useful to introduce the term $x$-intercept.

###### $x$-intercept

An $x$-intercept for a function $f(x)$ is the value of $x$ such that $f(x)=0\text{.}$

###### Note1.18

In the equation $f (x) = mx + b\text{,}$ we call $m$ and $b$ parameters. Their values are fixed for any particular linear equation; for example, in the equation $y = 2x + 3\text{,}$ $m = 2$ and $b = 3\text{,}$ and the variables are $x$ and $y\text{.}$ By changing the values of $m$ and $b\text{,}$ we can write the equation for any line except a vertical line (see Figure1.19). The collection of all linear functions $f (x) = mx + b$ is called a two-parameter family of functions.