The diagram shows the graph of $y = a \cos bx + c$, where $b 0$, for \(-2\pi \le

Question

The diagram shows the graph of $y = a \cos bx + c$, where $b > 0$, for $-2\pi \le x \le 2\pi$. The graph has a maximum point at $\left( \frac{\pi}{2}, 4 \right)$ and a minimum point at $(0, -8)$. Find the amplitude, the period and the range of values of $y$ corresponding to the interval. Hence find the values of $a$, $b$ and $c$. Write down the coordinates of the other maximum point(s) and the other minimum point(s) of the graph in the given interval. Find the range of the value of $k$ for which the equation $a \cos bx + c = k$ has more than four unique solutions for $-2\pi \le x \le 2\pi$.

Accepted Answer

(i) amplitude $6$, period $\pi $, range $-8 \le y \le 4, a = -6, b = 2, c = -2$; (ii) other maxima $\left(-\dfrac{3\pi}{2},4\right), \left(-\dfrac{\pi}{2},4\right), \left(\dfrac{3\pi}{2},4\right)$, other minima $(-2\pi ,-8), (-\pi ,-8), (\pi ,-8), (2\pi ,-8)$; (iii) $-8 \le k < 4$

Trigonometry - Trigonometric Graphs and Parameters

Trigonometry - Trigonometric Graphs and Parameters

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