Interpreting Functions Test Answer Key with Scoring – Algebra 2

Question 1 Rectangle area model (3 points)

1. The width and length 15 − w are positive: 0 < w < 15.
2. A = −w² + 15w = −(w − 7.5)² + 56.25. −(w − 7.5)² + 56.25.
3. Maximum 56.25 ft² at w = 7.5 ft.

Scoring: 1 point per part.

Question 2 Average rate of change (2 points)

k(1) = −2 and k(3) = 81 − 9 = 72. Rate = (72 − (−2))/(3 − 1) = 74/2 = 37.

Scoring: 1 point for both values, 1 point for the quotient.

Question 3 Rates from a table (2 points)

0 to 2: 18/2 = 9. 2 to 4: 32/2 = 16. 4 to 6: 46/2 = 23. 9, 16 and 23 ft/s: the drone is speeding up.

Scoring: 1 point for the three rates, 1 point for the trend.

Question 4 Zeros by completing the square (2 points)

(x − 4)² − 9 = 0, so x − 4 = ±3 and x = 1 or 7. The vertex is (4, −9). Zeros 1 and 7; vertex (4, −9).

Scoring: 1 point for vertex form, 1 point for the zeros.

Question 5 Features of a rational graph (3 points)

1. All real numbers except x = 1: x ≠ 1.
2. x = 1, where the denominator is 0.
3. As |x| grows, 2/(x − 1) → 0, so y = 3.

Scoring: 1 point per part.

Question 6 Compare a line and an exponential (3 points)

1. A: 3. B: (32 − 4)/3 = 28/3 ≈ 9.33.
2. A(5) = 25 and B(5) = 4 · 2⁵ = 128.
3. B, because it doubles each step (exponential growth), which eventually outpaces any constant rate.

Scoring: 1 point per part.

Question 7 Trig model features (2 points)

Period = 2π ÷ (π/4) = 8. Maximum = 20 + 5 = 25 and minimum = 20 − 5 = 15. Period 8; max 25; min 15.

Scoring: 1 point for the period, 1 point for the extremes.

Question 8 Value of a device (2 points)

V(2) = 1200(0.81) = 972. Rate = (972 − 1200)/2 = −114. The value falls by an average of $114 per year over the first two years.

Scoring: 1 point for the value, 1 point for the interpretation with units.

Question 9 Domain of a logarithm (1 point)

The argument must be positive, so x − 4 > 0. x > 4.

Scoring: 1 point.