Linear Functions Flashcards

Two lines are defined by y = 4x − 3 and y = (1/4)x + 2. Which statement about these lines is true?

Answer: They intersect at exactly one point

The slopes 4 and 1/4 are different (product = 1, not −1), so the lines are not perpendicular and not parallel — they intersect at exactly one point.

The function f(x) = −2x + 10 is defined for 0 ≤ x ≤ 5. What is the range of f?

Answer: 0 ≤ f(x) ≤ 10

At x = 0: f = 10; at x = 5: f = 0; since slope is negative, f decreases, so range is 0 ≤ f(x) ≤ 10.

The equation of a line is y = 5 − (3/2)x. What is the slope of a line perpendicular to this line?

Answer: 2/3

The slope of the given line is −3/2; the perpendicular slope is the negative reciprocal: 2/3.

A salesperson earns a base salary of $2,000 per month plus a 5% commission on all sales. If the salesperson earns $2,750 in a month, what were their total sales that month?

Answer: $15,000

2000 + 0.05s = 2750 → 0.05s = 750 → s = $15,000.

In the function f(x) = 6x + b, if the graph passes through (−2, 4), what is the value of b?

Answer: 16

4 = 6(−2) + b → 4 = −12 + b → b = 16.

A school's enrollment has been declining linearly. In 2020 the school had 840 students, and in 2024 it has 680 students. If this trend continues, in what year will enrollment reach 560 students?

Answer: 2027

Rate = (680 − 840)/4 = −40 students/yr; from 2024: 680 − 40t = 560 → 40t = 120 → t = 3 more years → 2027.

A linear function g satisfies g(2) = 7 and the slope is −4. What is g(5)?

Answer: −5

g(x) = −4x + b; 7 = −4(2) + b → b = 15; g(5) = −4(5) + 15 = −20 + 15 = −5.