Video Transcript
The lines on the following graph
show the amount of fuel remaining in the tank as a function of the distance covered
since the tank was 60 liters full for three different car models. Which of the following is the
function describing the quantity of fuel in the tank of the most environmentally
friendly car model represented in the above graph? Option (A) 𝑓 of 𝑥 equals 60 minus
three over 40 times 𝑥. Option (B) 𝑓 of 𝑥 equals 60 minus
one over 10 times 𝑥. Option (C) 𝑓 of 𝑥 equals 60 minus
one over 15 times 𝑥. Option (D) 𝑓 of 𝑥 equals one over
15 𝑥 minus 60. Or is it option (E) 𝑓 of 𝑥 equals
three over 40 𝑥 minus 60?
In this question, we are given a
graph showing the amount of fuel remaining in the tanks of three different models of
car as a function of the distance traveled. We want to use this graph to
identify the function describing the quantity of fuel remaining in the most
environmentally friendly car after it has traveled 𝑥 kilometers. To do this, let’s start by
considering which model of car will be the most environmentally friendly. We can assume that this will be the
car that travels the furthest distance on the 60 liters of fuel.
Since the 𝑥-coordinates of points
on the graph tell us the distance traveled and the 𝑦-coordinates tell us the amount
of fuel remaining, we can note that the 𝑥-intercepts will tell us the distance
traveled when there is no fuel remaining in the tank. We see that the black line has the
largest 𝑥-intercept at 900, so it is the most fuel-efficient car model in this
test. We want to find the function
representing the black line in this graph. We can achieve this by recalling
that a straight line has the equation 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is the slope
of the line and 𝑏 is its 𝑦-intercept. This is the same as saying that 𝑓
of 𝑥 equals 𝑚𝑥 plus 𝑏, so we just need to find the slope and 𝑦-intercept of the
black line in the graph.
We can start with the
𝑦-intercept. We see that all three lines share
the same 𝑦-intercept of 60, that is, the initial amount of fuel in the car
models. We can find the slope by recalling
it is the change in 𝑦 divided by the change in 𝑥. We see that as we travel between
the intercepts on the black line, the 𝑦-value decreases from 60 to zero and the
𝑥-values increase from zero to 900. Therefore, the slope is equal to
negative 60 over 900, which simplifies to give negative one over 15. If we substitute 𝑚 equals negative
one over 15 and 𝑏 equals 60 into the equation of a line, we obtain 𝑓 of 𝑥 equals
negative one over 15 𝑥 plus 60. We see that this is the same as
option (C) with the terms reversed.
