Given the function: What are the amplitude, period, and phase shift?

Question

BlackTom AI · Accepted Answer

To analyze this function, I’ll parse the standard form and extract its key parameters.
Option a states: Amplitude 3, Period π, Phase Shift -π/8.
Option b states: Amplitude 3, Period π, Phase Shift -π/4.
Option c states: Amplitude 2, Period π, Phase Shift -π/4.
Option d states: Amplitude 3, Period π, Phase Shift 0.

First, address amplitude: the given function is -3 cos(2t + π/4) - 5. The amplitude is the absolute value of the coefficient in front of the cosine term, which is |−3| = 3. So any correct choice must have Amplitude = 3. This immediately rules out option c, which lists Amplitude 2.

Next, determine the period: for a cosine function of the form cos(Bt + ...), the period is 2π divided by the absolute value of B. Here B = 2, so Period = 2π / 2 = π. Therefore, any correct choice must have Period = π. All remaining options a, b, and d fit this requirement, while c is already invalid.

Now, the phase shift: rewrite the inside as 2t + π/4 = 2(t + π/8). This shows the graph is shifted left by π/8 (since the argument is 2 times (t + π/8)). In standard form y = A cos(B(t − C)) + D, the phase shift is C. Here C = −π/8, representing a leftward shift of π/8. Thus the correct phase shift is −π/8.

With these three pieces combined, only option a has Amplitude = 3, Period = π, and Phase Shift = −π/8. Option b shares the correct amplitude and period but uses −π/4 for the phase shift, which is incorrect given the algebraic rewrite. Option d has the correct amplitude and period but zero phase shift, which would correspond to cos(2t) without the +π/4 term. Option c is invalid due to the incorrect amplitude.

Summary: matching amplitude 3, period π, and phase shift −π/8 is only consistent with option a.

MHF4U - Advanced Functions 12 (2025-26) - A

Given the function: What are the amplitude, period, and phase shift?

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