LEI - Modeling Linear Functions (Lesson)

Modeling Linear Functions

Let's revisit the phone plan with a cost of $0.15 per MB used.

Independent Variable: m MB's used	Dependent Variable: C(m) = 0.15m	(m, C(m))
1	C(1) = 0.15(1)= 0.15	(1, 0.15)
10	C(1) = 0.15(10)= 1.50	(10, 1.50)
25	C(1) = 0.15(25)= 3.75	(25, 3.75)
100	C(1) = 0.15(100)= 15	(100, 15)
180	C(1) = 0.15(180)= 27	(180, 27)

And let's look at the graphical representation of these points:

points graphed for the example problem

But, what if we used 50 MB of data? Or 121.5 MB of data? We need to consider this function as a continuous line so that we know the relationship between each amount of data used and the cost of our bill.

a line through the points representing the relationship

But first, we need to know how to graph lines!

Understanding Slope

Slope is the average rate of change of a function. For a line, the slope is considered the: rise/run. Slope = $LaTeX: \frac{rise}{run}=\frac{change\:in\:y}{change\:in\:x}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$ $\frac{rise}{run}=\frac{change\:in\:y}{change\:in\:x}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

We can also calculate slope algebraically using the formula: $LaTeX: m=\frac{y_2-y^{_{_1}}}{x_2-x_1}$ $m=\frac{y_2-y^{_{_1}}}{x_2-x_1}$

Example: Calculate the slope of the line that contains the points (1, -2) and (3, -5).

Let the first coordinate be x₁ and y₁. And let the second coordinate be x₂ and y₂_.
Substitute into the equation: $LaTeX: m=\frac{y_2-y_1}{x_2-x_1}=\frac{-5-\left(-2\right)}{3-1}=\frac{-5+2}{2}=\frac{-3}{2}$ $m=\frac{y_2-y_1}{x_2-x_1}=\frac{-5-\left(-2\right)}{3-1}=\frac{-5+2}{2}=\frac{-3}{2}$

So now we know our line has a negative slope which means it goes down from left to right. We also know two points on our line so we can graph it:

graph with a negative slope

Watch this video to practice a few more:

Slope Practice

What is the slope of each graph?

Find the slope of the line containing the given points.

(-2, 3) and (4, -1)
(3, -4) and (3, 5)
(5, -7) and (-5, -7)
(-1,-4) and (-4, 5)

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

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