The diagram shows the graph of $y = a \sin bx + c$, where $b 0$, for \(-180^\circ

Question

The diagram shows the graph of $y = a \sin bx + c$, where $b > 0$, for $-180^\circ \le x \le 180^\circ$. The graph has a maximum point at $\left( 45^\circ, \frac{9}{2} \right)$ and a minimum point at $\left( 135^\circ, -\frac{1}{2} \right)$. 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 $a \sin bx + c = k$ has four unique solutions for $-180^\circ \le x \le 180^\circ$.

Accepted Answer

(i) amplitude $= \dfrac{5}{2}$, period $= 180^\circ$, range $-\dfrac{1}{2} \le y \le \dfrac{9}{2}, a = \dfrac{5}{2}, b = 2, c = 2$; (ii) other maximum $\left(-135^\circ, \dfrac{9}{2}\right)$, other minimum $\left(-45^\circ, -\dfrac{1}{2}\right)$; (iii) $-\dfrac{1}{2} < k < 2$ or $2 < k < \dfrac{9}{2}$

Trigonometry - Trigonometric Graphs and Parameters

Trigonometry - Trigonometric Graphs and Parameters

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