The production of a certain car model dropped from seven million cars in 2001 to 3.85 million cars in 2012. Assuming the production decreased at a constant rate, which of the following linear functions 𝑓 best models the production, in millions of cars, 𝑡 years after 2001? A) 𝑓 of 𝑡 equals 77 over 220 times 𝑡 plus seven, B) 𝑓 of 𝑡 equals 63 over 220 times 𝑡 plus seven, C) 𝑓 of 𝑡 equals negative 77 over 220 times 𝑡 plus seven, or D) 𝑓 of 𝑡 equals negative 63 over 220 times 𝑡 plus seven.
The first thing we need to do is think about what we know about linear functions. A linear function, given in slope-intercept form, looks like this. 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑏 represents the 𝑦-intercept and 𝑚 represents the slope. And the slope is the changes in 𝑦 over the changes in 𝑥. If we think about modeling this situation on a graph, the 𝑥-axis would represent the years after 2001, the 𝑡 variable, and the 𝑦-axis would represent the number of cars produced in millions, 𝑓 of 𝑡.
We know that this graph should be going down and to the right. There should be a decrease. And this means we’re looking for a function that has a negative slope. Now we can eliminate options A and B as their slope values are positive. At this point, we’ll need to determine if it is a negative 77 over 220 slope or a negative 63 over 220 slope. Remember, we’re looking for the changes in 𝑦 over the changes in 𝑥. The cars produced went from seven to 3.85, so we’ll subtract seven minus 3.85. This is the change in cars produced. That’s the 𝑦 and the numerator.
We need to be careful as we calculate the changes in 𝑥. There were seven million cars in 2001, and there were 3.85 million cars in 2012. Notice that we’re subtracting 2012 from 2001. And that’s a difference of 11 years. And it’s going to be negative. 2001 minus 2012 is negative 11. And then seven minus 3.85 is 3.15. It’s also possible to do this the other way around. Take 3.85 and subtract seven and then subtract 2001 from 2012. In this case, you would get negative 3.15 over 11. And both of these values are equal to each other.
But now our challenge becomes, how do we take 3.15 over negative 11 and becomes something that’s recognizable from our answer choices? We can take 3.15 over 11 and try to give it a denominator of 220. We know that 11 times 20 equals 220. And if we multiply the denominator by a value, we need to multiply the numerator by the same value. And I’ll rewrite 3.15 times 20 as 3.15 times 10 times two. 3.15 times 10 equals 31 and a half. And 31 and a half times two equals 63. So, we have 63 over 220. But don’t forget, our original value is negative. We’re looking for a decrease in the production of cars. And so, we can say that 3.15 over negative 11 equals negative 63 over 220, which is option D.