Take this after Lesson 16. Half-length AP simulation covering Polynomial & Rational Functions (Unit 1) and Exponential & Logarithmic Functions (Unit 2).
| Section | Items | Time | Calculator |
|---|---|---|---|
| I, Part A | Questions 1–14 | 42 min | 🚫 NONE |
| I, Part B | Questions 15–20 | 18 min | 📱 graphing calculator |
| II | 2 FRQs (one 📱, one 🚫) | 30 min | as marked |
1. (A) [L1] f(5) = 5, f(1) = −3; (5 − (−3))/4 = 2.
2. (B) [L1/L2] Δ = 3, 5, 7: increasing outputs, increasing rates → increasing, concave up.
3. (C) [L4] Multiplicity 2 (even) → touch and turn.
lim x→−∞ p(x) =4. (D) [L4/L5] Degree 1 + 2 + 3 = 6 (even), leading coefficient positive → both ends +∞.
5. (A) [L6] (x + 3) cancels (equal multiplicity) → hole at −3; (x − 5) survives below → VA at 5.
6. (B) [L5] Equal degrees → ratio of leading coefficients: 3/4.
7. (C) [L7] C(4,2)·x²·(−3)² = 6·9 = 54. (A) forgets (−3)² is positive.
8. (A) [L8] x: solve x − 1 = 4 → 5; y: 2(−2) = −4 → (5, −4).
9. (D) [L2/L9] Constant second differences ⇔ quadratic.
10. (B) [L2/L8] Vertex form: minimum = −7 (at x = 3).
11. (A) [L10] 4·3⁴ = 324. (B) uses 3³; (C) 3⁵.
12. (B) [L11] 8^(x/3) = (8^(1/3))ˣ = 2ˣ.
13. (C) [L12] Keep 93%: b = 0.93.
14. (D) [L11] a = 5; 5b² = 45 → b² = 9 → b = 3.
15. (A) [L12] 1.18⁶ ≈ 2.700; 12.4·2.700 ≈ 33.5.
16. (B) [L16] (0.85)ᵗ = 0.3 → t = ln 0.3/ln 0.85 ≈ (−1.204)/(−0.163) ≈ 7.408.
17. (C) [L3] Local min at x = 3 (calculator minimum): f(3) = 81 − 108 = −27. (B) is f(2), not the min.
18. (B) [L12] Predicted: 3.2(1.4)⁴ = 3.2·3.8416 ≈ 12.293. Residual = 13.0 − 12.293 ≈ +0.71 (model underestimates). (A) is the flipped order.
19. (A) [L16] t = ln 2/ln 1.09 ≈ 0.693/0.0862 ≈ 8.04. (C) is the simple-interest answer.
20. (D) [L16] Linear on semi-log ⇔ exponential in raw form.
A podcast's subscriber count, in thousands, is recorded every two months:
| month t | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| S (thousands) | 40 | 52 | 67.6 | 87.9 | 114.2 |
(a) (i) Find the average rate of change of S over the interval [2, 6]. (ii) Interpret this value in context, with units.
(b) (i) Show that an exponential model is appropriate for these data, using ratios over equal-length intervals. (ii) Write an exponential model S(t) = a·bᵗ (you may leave b in exact exponent form, or use S(t) = a·r^(t/2) with the 2-month factor r).
(c) (i) Use your model to predict the subscriber count at t = 12 months. (ii) Give one reason this prediction should be treated with caution.
(a) Solve exactly: 4^(x+1) = 8ˣ.
(b) Solve, checking for extraneous solutions: log₃x + log₃(x − 6) = 3.
(c) The function f(x) = 2·10ˣ − 4. (i) Find f⁻¹(x). (ii) Evaluate f⁻¹(16).
1. (A) [L1] f(5) = 5, f(1) = −3; (5 − (−3))/4 = 2. 2. (B) [L1/L2] Δ = 3, 5, 7: increasing outputs, increasing rates → increasing, concave up. 3. (C) [L4] Multiplicity 2 (even) → touch and turn. 4. (D) [L4/L5] Degree 1 + 2 + 3 = 6 (even), leading coefficient positive → both ends +∞. 5. (A) [L6] (x + 3) cancels (equal multiplicity) → hole at −3; (x − 5) survives below → VA at 5. 6. (B) [L5] Equal degrees → ratio of leading coefficients: 3/4. 7. (C) [L7] C(4,2)·x²·(−3)² = 6·9 = 54. (A) forgets (−3)² is positive. 8. (A) [L8] x: solve x − 1 = 4 → 5; y: 2(−2) = −4 → (5, −4). 9. (D) [L2/L9] Constant second differences ⇔ quadratic. 10. (B) [L2/L8] Vertex form: minimum = −7 (at x = 3). 11. (A) [L10] 4·3⁴ = 324. (B) uses 3³; (C) 3⁵. 12. (B) [L11] 8^(x/3) = (8^(1/3))ˣ = 2ˣ. 13. (C) [L12] Keep 93%: b = 0.93. 14. (D) [L11] a = 5; 5b² = 45 → b² = 9 → b = 3. 15. (A) [L12] 1.18⁶ ≈ 2.700; 12.4·2.700 ≈ 33.5. 16. (B) [L16] (0.85)ᵗ = 0.3 → t = ln 0.3/ln 0.85 ≈ (−1.204)/(−0.163) ≈ 7.408. 17. (C) [L3] Local min at x = 3 (calculator minimum): f(3) = 81 − 108 = −27. (B) is f(2), not the min. 18. (B) [L12] Predicted: 3.2(1.4)⁴ = 3.2·3.8416 ≈ 12.293. Residual = 13.0 − 12.293 ≈ +0.71 (model underestimates). (A) is the flipped order. 19. (A) [L16] t = ln 2/ln 1.09 ≈ 0.693/0.0862 ≈ 8.04. (C) is the simple-interest answer. 20. (D) [L16] Linear on semi-log ⇔ exponential in raw form.
(a) (i) [1] (87.9 − 52)/(6 − 2) = 35.9/4 = 8.975 thousand subscribers per month. (ii) [1] From month 2 to month 6, the podcast gained subscribers at an average rate of ≈ 8,975 per month. (b) (i) [1] Ratios over each 2-month interval: 52/40 = 1.30, 67.6/52 = 1.30, 87.9/67.6 ≈ 1.300, 114.2/87.9 ≈ 1.299 — approximately constant factor 1.3 per 2 months → exponential growth. (Differences 12, 15.6, 20.3, 26.3 rule out linear.) (ii) [1] S(t) = 40·(1.3)^(t/2) (equivalently a ≈ 40, b = 1.3^(1/2) ≈ 1.140: S(t) ≈ 40·(1.140)ᵗ). (c) (i) [1] S(12) = 40·(1.3)⁶ = 40·4.8268 ≈ 193.1 thousand ≈ 193,000 subscribers. (ii) [1] Any one: t = 12 is outside the observed window [0, 8] (extrapolation); sustained 30%-per-2-month growth is unlikely as the audience saturates; the model assumes the growth rate never changes.
(a) [2] Common base 2: 2^(2(x+1)) = 2^(3x) [1] → 2x + 2 = 3x → x = 2 [1]. (Check: 4³ = 64 = 8² ✓) (b) [2] log₃(x(x − 6)) = 3 → x² − 6x = 27 → (x − 9)(x + 3) = 0 → x = 9 or x = −3 [1]; x = −3 makes both logs undefined — rejected. x = 9 only [1]. (Check: log₃9 + log₃3 = 2 + 1 = 3 ✓) (c) (i) [1] y = 2·10ˣ − 4 → 10ˣ = (y + 4)/2 → f⁻¹(x) = log((x + 4)/2). (ii) [1] f⁻¹(16) = log(20/2) = log 10 = 1.
Rough diagnostic bands (half-length test — treat as directional, not predictive): - ≥ 80% composite: on pace for a 5; proceed to Unit 3 at full speed - 65–79%: on pace for a 4; review flagged lessons before proceeding - 50–64%: on pace for a 3; redo the (e) problem sets of your three weakest lessons - < 50%: re-study the weakest unit's section (b) cores before continuing
| Question | Lesson | Topic |
|---|---|---|
| 1, 2 | L1–L2 | rates of change, concavity from tables |
| 3, 4 | L4–L5 | multiplicity, end behavior |
| 5, 6 | L5–L6 | rational functions: holes, VAs, HAs |
| 7 | L7 | binomial theorem |
| 8, 10 | L8 | transformations, vertex form |
| 9 | L2/L9 | model selection |
| 11 | L10 | geometric sequences |
| 12, 14 | L11 | exponential manipulation & construction |
| 13, 15, 18, 20 | L12/L16 | exponential models, residuals, semi-log |
| 16, 19 | L16 | solving exponential equations |
| 17 | L3 | extrema with the calculator |
| FRQ 1 | L9, L12 | modeling pipeline |
| FRQ 2 | L11, L14–L16 | symbolic manipulation suite |
Your running multiple-choice score appears in the bar below. Self-score the free-response section with the rubrics in the answer key, then use the diagnostic table to target review.