Which curve best represents the graph of the function $f(x) = -a \sin x + b \cos x$ given that the constants $a$ and $b$ are both positive?
A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2021 - Paper 1
Question 7
Which curve best represents the graph of the function $f(x) = -a \sin x + b \cos x$ given that the constants $a$ and $b$ are both positive?
A.
B.
C.
D.
Worked Solution & Example Answer:Which curve best represents the graph of the function $f(x) = -a \sin x + b \cos x$ given that the constants $a$ and $b$ are both positive?
A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2021 - Paper 1
Step 1
Identify the function behavior
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Answer
The given function is a combination of a negative sine function and a positive cosine function. The parameters a and b being positive suggest that the amplitude of the sine component is reflected downward, while the cosine component lifts the wave upward.
Step 2
Analyze the sine and cosine components
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Answer
Since −asinx contributes a downward shift and bcosx contributes an upward shift, the overall behavior of the function will exhibit periodic oscillation around the x-axis, but primarily dictated by the cosine curve's positive contribution.
Step 3
Select the correct graph
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Answer
The correct graph of the function will show periods where the graph remains positive above the x-axis and dips below according to the sine wave's influence. The curve that meets these criteria and represents the function accurately is D.
