The article titled “Slope” described real-world situations in which the concept of slope is useful. “Graphing Slope” is the next step. Moving the concept to graph paper or a graphing application.

Once you have a firm grasp of what slope is you can use it to help write equations that can answer questions. First you must understand the difference between a dependent and an independent variable.

In this example, I had two things that varied: the number of calories I was burning and the time I was running. When I graph these variables, I ask myself did the time I ran depend on the number of calories I burned, or did the calories I burned depend on how long I ran?

Because the calories burned depend on how long I run, the calories go along the y-axis and the time is on the x-axis.

In this second example, the calf’s weight depends on how long he has been growing. The weight belongs on the y-axis and the age of the calf lies along the x-axis.

Let’s revisit this example and write the given information as ordered pairs. Since the age of the calf is the independent variable, it is the x value of the ordered pair and the weight or dependent variable is the y value. Our two ordered pairs become (30, 190) and (45, 235). The rate of change or slope for this problem is the change in y (weight) over the change in x (age). When we found our rise over our run, we took our second y minus our first one (235 – 190) and found our weight change to be 45 pounds. Our run was the change in age (45 – 30) 15 days. Dividing 45 pounds of weight gain by 15 days, we learned the calf gained three pounds per day.

We can write an equation describing the calf’s growth as follows:

y = 3x + 100

Where did the 100 come from and what does it stand for? If we look at the graph of the line, it crosses the y-axis when y is 100. That’s the y-intercept. When x = 0, when the calf was born, it weighed 100 pounds. The neat thing about having the equation is that we can predict how much the calf will weigh when he is 3 months old.

y = 3(90) + 100

y = 370 pounds

We can also predict when he will be 750 pounds which is when many calves are sold to a feed lot.

750 = 3x + 100

x = 217 days or a little over 7 months

The length of the hair depends on how long she has been growing it so y corresponds to length and x corresponds to time in months. The rate of change or slope here is the ½ an inch per month. Let’s put the slope in the equation.

y = ½x + b

What is the y-intercept? If she starts growing her hair now, at time 0 months, how long is it?

y = ½x + 5

To find out how long it will take to grow the hair to 12 inches, substitute 12 in for y and solve for x.

Again, a strategy is to write these out as ordered pairs. Halloween represents day 0. (0, 258) November 5^{th} is six days of eating candy (6, 232). Calculate the slope. Write the equation in slope y-intercept form.

y = -13/3 x + 258

He will run out when there is no more candy left. That is the day when y = 0.

0 = -13/3 x + 258

Fill in the equation y = mx + b with the correct values for m and b.

y = 12x + 480

Substitute 2000 in for y and solve for x.

A slope foldable for note taking is available for download. Inspiration from ispeakmath.org.