NBT Mathematics Functions and Calculus 2 — Questions and Answers
Question 1: What is the maximum value of f(x) = −2x² + 8x − 3?
- 5 (Correct answer)
- 2
- 8
- −3
Correct answer: 5
The vertex is at x = −b/(2a) = 2; f(2) = −8 + 16 − 3 = 5.
Question 2: For f(x) = x³ − 3x, the x-values of the turning points are:
- x = ±1 (Correct answer)
- x = ±3
- x = 0 and x = 3
- x = ±√3
Correct answer: x = ±1
Setting f′(x) = 3x² − 3 = 0 gives x² = 1, so x = ±1.
Question 3: Which of the following equals ∫4x³ dx?
- x⁴ + C (Correct answer)
- 12x² + C
- 4x⁴ + C
- x⁴/4 + C
Correct answer: x⁴ + C
∫4x³ dx = 4 · x⁴/4 + C = x⁴ + C by the power rule for integration.
Question 4: The asymptotes of f(x) = 2/(x − 3) + 1 are:
- x = 3 and y = 1 (Correct answer)
- x = −3 and y = 1
- x = 3 and y = −1
- x = 1 and y = 3
Correct answer: x = 3 and y = 1
The vertical asymptote occurs where the denominator is zero (x = 3) and the horizontal asymptote is the constant being added (y = 1).
Question 5: The average rate of change of f(x) = x² from x = 1 to x = 3 is:
- 4 (Correct answer)
- 2
- 6
- 8
Correct answer: 4
[f(3) − f(1)] / (3 − 1) = (9 − 1) / 2 = 8/2 = 4.
Question 6: If f(x) = log₂(x), then f(8) equals:
- 3 (Correct answer)
- 2
- 4
- 16
Correct answer: 3
log₂(8) = 3 because 2³ = 8.
Question 7: The transformation that maps f(x) to f(x + 2) − 3 is:
- Shift left 2 and down 3 (Correct answer)
- Shift right 2 and down 3
- Shift left 2 and up 3
- Shift right 2 and up 3
Correct answer: Shift left 2 and down 3
Replacing x with x+2 shifts the graph 2 units left; subtracting 3 outside shifts it 3 units down.
NBT Mathematics Functions and Calculus 2
What is the maximum value of f(x) = −2x² + 8x − 3?