3.10 Function concepts -- lines

Most students entering Algebra II are already familiar with the basic mechanics of graphing lines. Recapping very briefly: the equation for a line is $y=\text{mx}+b$ where $b$ is the $y$ -intercept (the place where the line crosses the $y$ -axis) and m is the slope. If a linear equation is given in another form (for instance, $\mathrm{4x}+\mathrm{2y}=5$ ), the easiest way to graph it is to rewrite it in $y=\text{mx}+b$ form (in this case, $y=-\mathrm{2x}+2\frac{1}{2}$ ).

There are two purposes of reintroducing this material in Algebra II. The first is to frame the discussion as linear functions modeling behavior . The second is to deepen your understanding of the important concept of slope.

Consider the following examples. Sam is a salesman—he earns a commission for each sale. Alice is a technical support representative—she earns $100 each day. The chart below shows their bank accounts over the week.

Sam has some extremely good days (such as the first day, when he made $200) and some extremely bad days (such as the second day, when he made nothing). Alice makes exactly $100 every day.

Let d be the number of days, S be the number of dollars Sam has made, and A be the number of dollars Alice has made. Both S and A are functions of time. But $s(t)$ is not a linear function , and $A(t)$ is a linear function .

Linear Function

A function is said to be “linear” if every time the independent variable increases by 1, the dependent variable increases or decreases by the same amount .

Once you know that Alice’s bank account function is linear, there are only two things you need to know before you can predict her bank account on any given day.

• How much money she started with ($750 in this example). This is called the $y$ - intercept .
• How much she makes each day ($100 in this example). This is called the slope .

$y$ -intercept is relatively easy to understand. Verbally, it is where the function starts; graphically, it is where the line crosses the $y$ -axis.

But what about slope? One of the best ways to understand the idea of slope is to convince yourself that all of the following definitions of slope are actually the same.

So slope does not tell you where a graph is, but how quickly it is rising. Looking at a graph, you can get an approximate feeling for its slope without any numbers. Examples are given below.