PRE CALCULUS Pre-Calculus Polynomial Functions 5 — Questions and Answers
Question 1: Which transformation shifts f(x) = x³ down 4 units and left 2 units?
- g(x) = (x - 2)³ - 4
- g(x) = (x + 2)³ - 4 (Correct answer)
- g(x) = (x - 2)³ + 4
- g(x) = (x + 2)³ + 4
Correct answer: g(x) = (x + 2)³ - 4
Replacing x with (x + 2) shifts left 2 and subtracting 4 shifts down 4.
Question 2: Which correctly describes the intermediate value theorem for polynomials?
- If f(a) > 0 and f(b) > 0, then f has a zero between a and b
- If f(a) and f(b) have opposite signs, then f has at least one zero between a and b (Correct answer)
- Every polynomial has a zero between any two real numbers
- Polynomials are always positive between consecutive zeros
Correct answer: If f(a) and f(b) have opposite signs, then f has at least one zero between a and b
The IVT guarantees a zero exists between a and b when f(a) and f(b) have opposite signs.
Question 3: What is the local maximum value of f(x) = -x³ + 3x near x = -1?
- -2
- 2 (Correct answer)
- 0
- -4
Correct answer: 2
f(-1) = -(-1)³ + 3(-1) = 1 - 3 = -2; checking x = 1: f(1) = -1 + 3 = 2, which is the local max.
Question 4: For f(x) = 2(x - 3)²(x + 1), the graph touches (but does not cross) the x-axis at which x-value?
- x = -1
- x = 3 (Correct answer)
- x = 0
- x = 2
Correct answer: x = 3
The factor (x - 3)² has even multiplicity 2, so the graph touches at x = 3 without crossing.
Question 5: Which polynomial has end behavior: as x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞?
- f(x) = 3x⁴ + 1
- f(x) = -x³ + 2x (Correct answer)
- f(x) = x⁵ - 4
- f(x) = -2x² + x
Correct answer: f(x) = -x³ + 2x
Odd degree with negative leading coefficient gives opposite-direction end behavior: up left, down right.
Question 6: A cubic polynomial with positive leading coefficient has which end behavior?
- Down left, down right
- Up left, up right
- Down left, up right (Correct answer)
- Up left, down right
Correct answer: Down left, up right
Odd degree with positive leading coefficient: falls to -∞ on the left and rises to +∞ on the right.
Question 7: If synthetic division of f(x) by (x - 3) gives a remainder of 0, which conclusion is valid?
- x = 0 is a zero of f
- f(3) = 3
- (x - 3) is a factor of f(x) (Correct answer)
- f has no other real zeros
Correct answer: (x - 3) is a factor of f(x)
A remainder of 0 means (x - 3) divides evenly into f(x), so it is a factor.
PRE CALCULUS Pre-Calculus Polynomial Functions 5
Which transformation shifts f(x) = x³ down 4 units and left 2 units?