Which curve best represents the graph of the function $f(x) = -a \, ext{sin} \, x + b \, ext{cos} \, x$ given that the constants $a$ and $b$ are both positive? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2021 - Paper 1

To determine the correct graph, we first analyze the function. The function is composed of a negative sine term and a positive cosine term, given that constants $a$ and $b$ are both positive.

1. Effect of $-a \, ext{sin} \, x$: The negative sine component means the wave will be flipped vertically. This will create a graph that starts high at $x=0$ (since $ext{sin}(0) = 0$ which contributes 0) and dips down from here.

2. Effect of $b \, ext{cos} \, x$: The positive cosine component will ensure that the graph has a peak, rather than a trough, as $x$ increases. The cosine function begins at its maximum value, which, when weighted by a positive $b$, will also be positive.

Given the combination of the two terms, we expect the graph to start high, dip down, and then rise again, creating a wavelike pattern with peaks and troughs.

3. Choosing from the options: We examine each of the provided option graphs:
   • Curve A: Starts low, not favorable.
   • Curve B: Has a symmetrical pattern not consistent with the negative sine component.
   • Curve C: Begins low and does not depict the required peaks.
   • Curve D: Starts at the highest point, dips to a minimum, aligns with expectations of both terms consistently interacting, and agrees with the behavior described.

Thus, the correct graph that represents the function is D.