Rise Placement Test Functions and Quadratic Equations 5 — Questions and Answers
Question 1: What is the minimum value of f(x) = x² − 6x + 10?
- 1 (Correct answer)
- 10
- 3
- −6
Correct answer: 1
Completing the square: f(x) = (x−3)² + 1, so the minimum value is 1 at x = 3.
Question 2: Which equation has roots x = 4 and x = −1?
- x² − 3x − 4 = 0 (Correct answer)
- x² + 3x − 4 = 0
- x² − 5x + 4 = 0
- x² + 5x − 4 = 0
Correct answer: x² − 3x − 4 = 0
(x−4)(x+1) = x² − 3x − 4 = 0 has roots x = 4 and x = −1.
Question 3: A function f is defined as f(x) = x² for x ≥ 0. What is its inverse function?
- f⁻¹(x) = x²
- f⁻¹(x) = −√x
- f⁻¹(x) = √x (Correct answer)
- f⁻¹(x) = 1/x²
Correct answer: f⁻¹(x) = √x
Swapping x and y in y = x² gives x = y², so y = √x for x ≥ 0.
Question 4: Completing the square on x² + 8x gives:
- (x + 4)² − 16 (Correct answer)
- (x + 4)² + 16
- (x + 8)² − 64
- (x − 4)² + 16
Correct answer: (x + 4)² − 16
x² + 8x = (x+4)² − 16 by adding and subtracting (8/2)² = 16.
Question 5: The height of a projectile is given by h = −5t² + 20t + 25. What is the initial height (t = 0)?
- 20 m
- 25 m (Correct answer)
- 5 m
- 45 m
Correct answer: 25 m
At t = 0: h = −5(0) + 20(0) + 25 = 25 meters.
Question 6: If f(x) = x² + 1 and g(x) = 2x − 3, what is (f ∘ g)(1)?
- 2 (Correct answer)
- 0
- −1
- 2
Correct answer: 2
g(1) = 2(1) − 3 = −1, then f(−1) = (−1)² + 1 = 2.
Question 7: For the equation y = 2(x − 1)² − 8, what are the x-intercepts?
- x = 1 ± 2
- x = 1 ± 4
- x = 3 and x = −1 (Correct answer)
- x = 5 and x = −3
Correct answer: x = 3 and x = −1
Setting y = 0: 2(x−1)² = 8 → (x−1)² = 4 → x − 1 = ±2 → x = 3 or x = −1.
Rise Placement Test Functions and Quadratic Equations 5
What is the minimum value of f(x) = x² − 6x + 10?