College Algebra

Example 6.22.

Market researchers are trying to understand the demand for fidget spinners among college students. Their analysis of the data they collected estimates that at a price of \(x\) dollars, the number of fidget spinners that will be bought is given by the function \(d(x)=-13x+572\text{.}\) We want to answer the following questions:

1. As the price increases, what happens to demand (the number of fidget spinners purchased)?

2. If they give away the spinners for free, how many should they expect students will take?

We can answer these questions by understanding what the numbers in the formula represent. Let's do them one at a time.

1. As the price increases, what happens to demand (the number of fidget spinners purchased)? Since this is about how the outputs (number of fidget spinners sold, the y-values) change as we change the inputs (the price, the x-values), we need to look at slope. The slope of this linear function is negative, which means it is decreasing. As we increase the price of the fidget spinners, the number sold will decrease.

2. If they give away the spinners for free, how many should they expect students will take? Since this is about what the output is (the number of fidget spinners, the y-value) when the input is 0 (the price, the x-value), we need to look at the y-intercept. In this function, the y-intercept is 572. So, if they are given away for free, students will take 572 of them, according to this model.

Example 6.24.

Economists are looking at how the price of different goods and services have changed since 2010. The functions they found from their data models are given in the table below.

We want to answer the following questions:

1. Which products have gotten more expensive over time, and which have gotten cheaper?

2. Which product has increased in price most quickly?

3. Which product was the most expensive in 2010?

For each of these, we will need to look at the slope and the y-intercept, so let's isolate those and add them to our table:

Now we are ready to answer each question one at a time.

1. Which products have gotten more expensive over time, and which have gotten cheaper? This is a question about how the output (price) changes based on the input (time), which is slope. Getting more expensive would mean that the price is increasing over time, so the slope is positive. Getting cheaper means that the price is decreasing over time, so the slope is negative. Therefore, the toothpaste, pizza, and movie tickets have gotten more expensive and the watermelon and plastic silverware have gotten cheaper.

2. Which product has increased in price most quickly? We already figured out that the slope is how we determine whether the price is increasing or decreasing. Since we want to know which is increasing quickest, we want to look for the highest slope. So, the movie tickets are increasing most quickly.

3. Which product was the most expensive in 2010? In the question, we were told that the models start in 2010, which means that the year 2010 is when \(x=0\text{.}\) So, we want to know the highest output (price) when \(x=0\text{,}\) which is exactly the y-intercept. Looking at that column of the table, we see that the movie tickets were the most expensive in 2010.