The table shows the fare F(d), in dollars, for a taxi ride of... | SAT Linear functions Question, Answer Explained

Math · AlgebraMedium

Distance, $d$ (miles)	Fare, $F(d)$ (dollars)
2	9.50
5	17.00
8	24.50

The table shows the fare $F(d)$ , in dollars, for a taxi ride of $d$ miles. The fare is a linear function of the distance. Which equation defines $F$ ?

$F(d) = 2.5d + 4.5$

$F(d) = 2.5d + 9.5$

$F(d) = 4.75d$

$F(d) = 7.5d + 4.5$

Show answer and explanation

Answer: A

From $d = 2$ to $d = 5$ , the fare rises by $17.00 - 9.50 = 7.50$ dollars over $3$ miles, so the slope is $\frac{7.5}{3} = 2.5$ . Then $9.50 = 2.5(2) + b$ gives $b = 4.5$ . Check the last row: $F(8) = 2.5(8) + 4.5 = 24.5$ .

Choice D uses $7.5$ , the change in fare from one row to the next, as the slope, but the rows are $3$ miles apart.

Why the other choices are wrong

B. This uses the first fare, 9.50, as the y-intercept, but 9.50 is the fare for 2 miles, not 0 miles.
C. This divides 9.50 by 2, which assumes the fare is proportional to distance. It gives F(5) = 23.75, not 17.00.
D. 7.5 is the change in fare over 3 miles. The slope is the change per mile, 2.5.

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