Video Transcript
A bus service charges a fixed fee of five pounds and an additional two pounds for every bus stop passed. Write a polynomial function to represent the total cost of a ride.
First, we remember that a polynomial function is a function that’s a sum of monomial terms, and monomial terms are terms containing constants, variables, and nonnegative integer exponents. This means we need to find a way to represent the cost of this bus ride using variables, constants, and sums. To calculate this, let’s imagine that we were going to travel one stop. The cost of this trip would be the five pounds’ fixed rate plus two pounds for traveling one stop. That’s two times one. Traveling one stop is then equal to seven pounds.
What if we travel two stops? Again, we would need to pay the five-pound fixed fee. But this time we need to multiply two by two. We’re traveling two stops, and therefore we’ll pay an additional two pounds. The total of nine pounds comes from the five-pound fixed fee and four pounds, two for each of the stops. Now let’s imagine that we traveled 10 stops on the bus. It’s a five-pound fixed fee plus two times 10. Traveling 10 stops would cost 25 pounds.
But how do we describe this using a function? In other words, we need an equation that describes the cost given any number of stops. We can then define the number of bus stops passed as 𝑥 stops. No matter what, we’ll pay this fixed fee of five pounds. And then we would need to multiply two times 𝑥 stops. Therefore, the cost is equal to five plus two 𝑥. We’ll let 𝑓 of 𝑥 be the total cost in pounds for traveling on the bus. Then, since it’s most usual to write the highest power of 𝑥 first, we can define 𝑓 of 𝑥 to be equal to two 𝑥 plus five. The total cost in pounds of a bus service for 𝑥 number of stops passed is 𝑓 of 𝑥 equals two 𝑥 plus five.
