PRE CALCULUS Pre-Calculus Polynomial Functions 5

Question 1

Which transformation shifts f(x) = x³ down 4 units and left 2 units?

Accepted Answer

g(x) = (x + 2)³ - 4

Answer

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?

Accepted Answer

If f(a) and f(b) have opposite signs, then f has at least one zero between a and b

Answer

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?

Accepted Answer

2

Answer

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?

Accepted Answer

x = 3

Answer

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) → ∞?

Accepted Answer

f(x) = -x³ + 2x

Answer

Odd degree with negative leading coefficient gives opposite-direction end behavior: up left, down right.

PRE CALCULUS Practice Test