How Do I Deal With Variables Other Than x in Linear Functions?

When working with functions with unusual variable names and scaling, students often run into two problems.

  • trouble with what each variable represents
  • trouble with the scaling of the variables

Since we rarely use x in Finite Math to represent the independent variable, we need to get used to these names. Typically, the variables are aligned or scaled. This means we need to pay particular care to interpreting answers.

In one type of the problem, you are given a demand function $latex \displaystyle p=D(q)=32-1.25q$ where p is the price in dollars and q is the quantity of watches demanded in hundreds. In part b, you were asked to find the price when the demand is 800 watches. This quantity corresponds to $latex \displaystyle q=8$ and gives a price of

$latex \displaystyle D(8)=32-1.25(8)=22$

Notice how the scaling on the variable works…800 watches is input as 8 hundred. In part c, you need to find the demand when the price is $27. Now we know the output from the function and we need to find the corresponding input,

$latex \displaystyle 32-1.25q=27$

When we solve this for q we get $latex \displaystyle q=4$. In other words, 4 hundred watches are demanded when the price is $27. Since the units on the answer blank is watches, you will need to type 400 in the answer blank. I think many of you were not taking the units into account on this problem. Units are incredibly important in the real world where you are dealing with millions of dollars. In those cases misinterpreting $2,000,000 and 2 million dollars may cost you your job.

How Do I Use WolframAlpha To Graph a Formula?

A number of you have asked me about alternatives to a TI graphing calculator. In addition to Excel, there are many online graphing tools available. If you have an Android phone or IPhone, there are a huge number of free apps that are available. I have had a hard time finding just one that does everything we need. Most will graph formulas, but may or may not graph data and fit data to lines. None of them work identically to a graphing calculator which makes them difficult to support. But luckily there is another option!

Another option is the website WolframAlpha (http://www.wolframalpha.com/). This website is the Internet’s leading computational engine. It can do just about anything. The trouble is knowing how to use it to do just about anything.

When you go to this website, you’ll see a box in which you can enter commands to help you do mathematics. The list of things you can do is HUGE. It is best to show some examples to get you started.

For instance, suppose you want to graph the function y = -5.686x + 676.173 in a window from x = 0 to x = 25. Enter the text you see below into WolframAlpha followed by Enter.

WolframAlpha makes a nice graph. If you want to graph two formulas simultaneously, add another formula with the word “and” as you see below.

You can see that these lines are going to meet. To see this point of intersection, extend the graph by modifying the input to WolframAlpha as you see below.

We can find this point of intersection by modifying the input to WolframAlpha with the command “intersections”:

Not only does it give a decimal…it also gives the exact answer in terms of fractions!

Often you are interested in evaluating the model above at a particular point. We can do this by clicking on the formula above. This will make WolframAlpha graph the formula by itself. Now add “where x = 20” on the command line and you will see:

This gives the same output as TRACE on a graphing calculator. I’ll continue to make posts about WolframAlpha so those of you who do not have access to a graphing calculator have an alternative way to make graphs, calculate values, ect.

How Do You Find The Intercepts of a Linear Function?

Problem 2 The percent p of high school seniors who ever used marijuana can be related to x, the number of year after 2000, by the equation 25p + 21x = 1215.

a. Find the x intercepts of the graph of this function.

b. Find and interpret the p intercept of the graph of this function.

c. Graph the function using the intercept.

How Is The Slope Of A Function Interpreted?

The slope is best understood by looking at the units of the independent and dependent variable. The units are always the units on the dependent variable per the units on the independent variable. This problem illustrates this fact.

Problem 1 The percent p of high school seniors who ever used marijuana can be related to x, the number of year after 2000, by the equation 25p + 21x = 1215.

a. Use this model to determine the slope of the graph of this function if x is the independent variable.

b. What is the rate of change of the percent of high school seniors using marijuana per year?

 

This last part indicates that each year, the percentage drops by 0.84 percent.