MLF - Modeling Linear Functions (Lesson)

Modeling Linear Functions

We use functions to tell us about relationships between values. For instance, let's say your cell phone plan charges you $0.15 per MB of data used. So we can write a function for the cost (C) in terms of the amount of data used. We would say LaTeX: C(m) = 0.15mC(m)=0.15m.

input m goes into the rule c of m equals 0.15m equals output

Let's take a look at a table of values for this function. 

Data Used (MBs)

Independent Variable:

m

Cost of Plan

Dependent Variable:

C(m) = 0.15m

Coordinates

(m, C(m))
1

LaTeX: C(1) = 0.15(1)= 0.15C(1)=0.15(1)=0.15

(1, 0.15)
10

LaTeX: C(10)=0.15(10)=1.50C(10)=0.15(10)=1.50

(10, 1.50)
25

LaTeX: C(25)=0.15(25)=3.75C(25)=0.15(25)=3.75

(25, 3.75)
100

LaTeX: C(100)=0.15(100)=15C(100)=0.15(100)=15

(100, 15)
180

LaTeX: C(180)=0.15(180)=27C(180)=0.15(180)=27

(180, 27)

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

1_Graph1.png

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.

1_Graph2.png

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}riserun=changeinychangeinx=ΔyΔx=y2−y1x2−x1 

Note: The Greek symbol LaTeX: \DeltaΔ (delta) stands for "change in" so LaTeX: \Delta yΔy would be interpreted as "change in y."

We can also calculate slope algebraically using the formula: LaTeX: m=\frac{y_2-y^{_{_1}}}{x_2-x_1}m=y2−y1x2−x1 

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

  1. Let the first coordinate be x1 and y1. And let the second coordinate be x2 and y2.   (x1, y1) = (1, -2) and (x2, y2) = (3, -5) 
  2. 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=y2−y1x2−x1=−5−(−2)3−1=−5+22=−32 

This line has a negative slope which means it decreases (goes down from left to right). Since we also know two points on our line, we can graph it:

Plot both points and use the slope (right 2 and down 3) to plot more points. 

1_Graph3.jpg

Connect the points to graph the line!

1_Graph4.jpg

Watch this video to practice a few more:

 

Slope Practice

What is the slope of each graph?

1. Slope1.png                                                     2. Slope2.png 

3. Slope3.png                                                      4.  Slope4.png 

Find the slope of the line containing the given points.

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

6. (3, -4) and (3, 5)

7. (5, -7) and (-5, -7)

8. (-1,-4) and (-4, 5)

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

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