The graph $y = a + b\cos(cx)$ is defined for $0 \le x \le 2\pi$, where $a$, $b$ an

Question

The graph $y = a + b\cos(cx)$ is defined for $0 \le x \le 2\pi$, where $a$, $b$ and $c$ are constants. The graph has a period of $\frac{4\pi}{3}$ and a minimum value of $1$ when $x = \frac{4\pi}{3}$. The graph also passes through the point $\left(\frac{\pi}{3}, 3\right)$. Show that $a = 3$, $b = -2$ and $c = \frac{3}{2}$.[3] Sketch the graph of $y = a + b\cos(cx)$ for $0 \le x \le 2\pi$.[2] Given that there are exactly $3$ solutions for the equation $b\cos(cx) = k - a$, for $0 \le x \le 2\pi$, state the range of values of the constant $k$.[1]

Accepted Answer

(a) $a=3,\ b=-2$; the positive-frequency convention gives $c=\dfrac32$, while $c=-\dfrac32$ gives the same graph. (b) $y=3-2\cos\left(\dfrac{3x}2\right)$ on $[0,2\pi]$, with extrema $1,5$. (c) $1<k<5$.

2024 CGSS P1 Q6

2024 CGSS P1 Q6

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