Linear functions arise naturally whenever we think of constant growth. In this section we’ll think about another very common type of function which arises naturally whenever we think of percent growth.
Percents are always of something — they are a part of some whole. For example, in a previous section, a report said that the average price of gasoline in Minnesota increased approximately 20% per year. If we were being very careful, we would say that this is 20% of the current price each year.
In a recent game against the Golden State Valkyries, Minnesota Lynx star Napheesa Collier’s field goal percentage was 72.7%. This means that she made 72.7% of .
Jocelyn got a job right out of college, as an administrative assistant earning $28,000 a year. The position turned out to be a great fit for her, and after one year she was promoted to data analyst with a 15% raise. The next year Joceyln was promoted again, to senior data analyst along with a 21% raise. “Not bad,” her friend Audun said, “a 36% raise in two years.” But Jocelyn quickly corrected him. “Audun, it’s even better than that! It’s over 39%!”
After the first year, Jocelyn’s salary of $28,000 was increased by 15%. Remember that means that her raise was 15% of $28,000. To calculate 15% of $28,000 we multiply using the decimal form
After the second year, Jocelyn got a 21% raise. But 21% of what? This means her salary rose by 21% from what it was just before the raise, that is, from the $32,200. (The 21% does not refer back to the original $28,000 value!) So, to calculate the second increase, we take 21% of $32,200, which is
What’s going on here? Audun thought that 15% and 21% would be 36% because \(15 + 21 = 36\text{.}\) The reason it doesn’t work that way is that while the 15% is of the original $28,000, the 21% is of the $32,200. So, we can’t just combine percentages by adding, because the wholes might be different.
SubsectionA one-step process for finding percent change
Each time we figured out Jocelyn’s new salary, we did a two-step process. First, we calculated the amount of the increase, and second, we found the new value by adding on. There’s actually an easier way.
Jocelyn’s salary was $28,000 and then went up by 15%. For her new salary we want to add her old salary (all of it) plus 15%. So we want 100% of her old salary plus 15% of her old salary. Since we are finding two percents of the same whole, we can add the percents together: in this case, we can compute 115% of her old salary.
That works in general. When we increase a number by 15%, we end up with 100% of what we started with plus 15% more of what we started with, for a grand total of 115% of what we started with. So we can just multiply by 1.15, which is 115% written in decimal, since
Now hang on to your hat, because we can combine these parts together. In our example, we started with $28,000. Then we multiplied by 1.15, which gave us $32,200. And then we multiplied that answer by 1.21, to get our final answer of $28,962. So really we just did
The comprehensive fee at a local private college is $64,000. The fee is projected to increase by 5.8% per year. Let’s test that the two-step method and the one-step method give the same result for the fee next year.
Here’s a little vocabulary that will be very helpful for us moving forward. A percentage increase is known as the growth rate and the number we multiply in the one-step method is called the growth factor.
Aside
For example, in calculating 15% increase, the growth rate was \(15\%=0.15\) in decimal, and so the growth factor was
The comprehensive fee at a local private college is $64,000. The fee is projected to increase by 5.8% per year. Let’s identify the growth rate and the growth factor.
SubsectionUsing exponents to find multiple increases at once
Jocelyn’s most recent assignment has been analyzing information on rising health care costs. In 2007 the United States spent $2.26 trillion on health care. Written out with all its zeros that’s
So, to find the effect of a 6.7% increase, we can just multiply by 1.067. Again, that’s the 100% of what we started with plus 6.7% more for a grand total of \(106.7\%= 1.067\text{.}\)
I don’t know about you, but I would rather not type that all into a calculator — I’m worried that I would lose count of how many times I typed 1.067. Luckily, multiplying by 1.067 ten times is the same as multiplying by \(1.067^{10}\text{.}\) Recall that the base \(1.067\) is the number we multiply by and the exponent (or power) \(10\) tells us how many times. So we can calculate
Notice that the order of operations is exactly what we wanted here: first raise 1.067 to the 10th power, then multiply by 2.26. So we can enter it all at once without needing parentheses. Bottom line: health care costs are expected to be around $4.32 trillion by the year 2017. Oh my!
The comprehensive fee at a local private college is $64,000. The fee is projected to increase by 5.8% per year. Let’s use exponents to compute some projected future fees.
We’re so close to the equation for Jocelyn’s health care costs assignment now, we can smell it. Our variables are
\begin{align*}
H \amp= \text{ health care costs (\$ trillions) } \sim \text{ dep} \\
Y \amp= \text{ time (years since 2007) } \sim \text{ indep}
\end{align*}
The comprehensive fee at a local private college is $64,000. The fee is projected to increase by 5.8% per year. Let’s write an equation that describes this story.
\begin{equation*}
H = 2.26\ast 1.067^{Y}
\end{equation*}
This type of equation is called an exponential equation because the independent variable is in the exponent. Any exponential equation fits this template.
Notice our two variables are in our equation and there are two constants. Each constant has its own meaning. The first constant is 2.26 and it is measured in trillions of dollars. It is the amount spent on health care in the starting year of 2007. In our standard form we refer to this quantity as the starting value (or start for short). As with linear equations, its official name is intercept and it’s the value where the curve crosses the vertical axis on the graph.
The second constant is the growth factor 1.067, and is the decimal equivalent of the 106.7%. The growth factor for an exponential equation is similar to the slope of a linear equation because both tell us how fast the function is increasing. But the slope measures the rate of change — how much is added at each step — while the growth factor corresponds to the percent increase. Another way to say that is linear functions correspond to situations where we are adding the same amount each time and exponential functions correspond to situations where we are adding the same percentage each time (or, equivalently, multiplying by the same amount each time).
Sometimes the graph of an exponential equation looks a lot like a straight line, especially if you only plot a few points. A good rule of thumb is to plot five or more points to see the curve in the graph of an exponential equation.
They eat the tops of my tulips in early spring and my lilies all summer long. Back in 2007 there were an estimated 1,800 rabbits in my neighborhood. Rabbits multiply quickly, 13% per year by one estimate.
What is a realistic domain? That means, for how many days do you think this model is reasonable? To keep a sense of scale, there are 1,094 students currently living in the dorms.
Her colleague Tomáš started with a salary of $78,000 but did not get a raise the first year like Mai did. What percentage raise would Tomáš need now in order to have the same final salary as Mai?
