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Lines

Module by: Kenny M. Felder. E-mail the author

Summary: This module discusses lines and their uses, and slope.

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=mx+by=mx+b size 12{y= ital "mx"+b} {} where bb size 12{b} {} is the yy size 12{y} {}-intercept (the place where the line crosses the yy size 12{y} {}-axis) and m is the slope. If a linear equation is given in another form (for instance, 4x+2y=54x+2y=5 size 12{4x+2y=5} {}), the easiest way to graph it is to rewrite it in y=mx+by=mx+b size 12{y= ital "mx"+b} {} form (in this case, y=−2x+212y=−2x+212 size 12{y= - 2x+2 { { size 8{1} } over { size 8{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.

Table 1
After this many days (t) Sam’s bank account (S) Alice’s bank account (A)
0 (*what they started with) $75 $750
1 $275 $850
2 $375 $950
3 $450 $1,050
4 $480 $1,150
5 $530 $1,250

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)s(t) size 12{s \( t \) } {} is not a linear function, and A(t)A(t) size 12{A \( t \) } {}is a linear function.

Definition 1: 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 yy size 12{y} {}-intercept.
  • How much she makes each day ($100 in this example). This is called the slope.

yy size 12{y} {}-intercept is relatively easy to understand. Verbally, it is where the function starts; graphically, it is where the line crosses the yy size 12{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.

Table 2
Definitions of Slope
In our example In general On a graph
Each day, Alice’s bank account increases by 100. So the slope is 100. Each time the independent variable increases by 1, the dependent variable increases by the slope. Each time you move to the right by 1, the graph goes up by the slope.
Between days 2 and 5, Alice earns $300 in 3 days. 300/3=100.Between days 1 and 3, she earns $200 in 2 days. 200/2=100. Take any two points. The change in the dependent variable, divided by the change in the independent variable, is the slope. Take any two points. The change in yy size 12{y} {} divided by the change in xx size 12{x} {} is the slope. This is often written as ΔyΔxΔyΔx size 12{ { {Δy} over {Δx} } } {}, or as riserunriserun size 12{ { { ital "rise"} over { ital "run"} } } {}
The higher the slope, the faster Alice is making moey. The higher the slope, the faster the dependent variable increases. The higher the slope, the faster the graph rises as you move to the right.

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.

Figure 1
(a) A slope of 1: each time you go over 1, you also go up 1(b) A steep slope of perhaps 3 or 4(c) A gentle slope of perhaps 1212.(d) A horizontal line has a slope of 0: each time you go over 1, you don’t go up at all!(e) This goes down as you move left to right. So the slope is negative. It is steep: maybe a –2.
A Line with a positive slope of 1A Line with a sttep positive slope of about 3 or 4A Line with a gentle positive slope of about 1/2A horizontal line with a no slopeA Line with a steep negative slope of about -2

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