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

The presence of the $-a \sin x$ term indicates that the function will oscillate in a way that the sine component is inverted, affecting the overall shape of the curve. The positive $b \cos x$ term will contribute positively to the graph.

At points where oldsymbol{x = 0}, $f(0) = b > 0$ because $a$ and $b$ are positive. This means the graph starts above the x-axis. As $x$ increases, the graph will oscillate, reflecting the negative sine component and the positive cosine component.

By analyzing the graphs presented in options A, B, C, and D, we want to find a graph that indicates the behavior of oscillating around the x-axis with an initial upward shift due to the positive value at $x = 0$.

The correct choice, based on the expected behavior of the graph of the function, is option D.