Question

graph $h(x) = 2\sin(\pi x - 3\pi) - 4$ in the interactive widget. note that one moveable point always defines an extremum point in the graph and the other point always defines a neighbouring intersection with the midline.

Explanation:

Step1: Simplify the sine argument

Use sine identity $\sin(A-B)=\sin A\cos B-\cos A\sin B$, and $\cos3\pi=-1$, $\sin3\pi=0$:
$$\sin(\pi x - 3\pi)=\sin(\pi x)\cos(3\pi)-\cos(\pi x)\sin(3\pi)=-\sin(\pi x)$$
Rewrite the function:
$$h(x)=2(-\sin(\pi x)) - 4=-2\sin(\pi x)-4$$

Step2: Identify key wave parameters

• Amplitude: $|A|=2$
• Midline: $y=-4$
• Period: $T=\frac{2\pi}{\pi}=2$

Step3: Find extremum point

Maximum of $h(x)$: $-4 + 2=-2$; solve $-\sin(\pi x)=1\Rightarrow \sin(\pi x)=-1\Rightarrow \pi x=\frac{3\pi}{2}+2k\pi\Rightarrow x=\frac{3}{2}+2k$. Take $k=1$, $x=\frac{7}{2}=3.5$, so extremum is $(3.5, -2)$
Minimum of $h(x)$: $-4 -2=-6$; solve $-\sin(\pi x)=-1\Rightarrow \sin(\pi x)=1\Rightarrow \pi x=\frac{\pi}{2}+2k\pi\Rightarrow x=\frac{1}{2}+2k$. Take $k=1$, $x=\frac{5}{2}=2.5$, so extremum is $(2.5, -6)$

Step4: Find midline intersection

Solve $h(x)=-4$:
$$-2\sin(\pi x)-4=-4\Rightarrow \sin(\pi x)=0\Rightarrow \pi x=k\pi\Rightarrow x=k$$
Take $k=3$, intersection point is $(3, -4)$

Answer:

1. Extremum point example: $\boldsymbol{(3.5, -2)}$ (local maximum) or $\boldsymbol{(2.5, -6)}$ (local minimum)
2. Neighbouring midline intersection: $\boldsymbol{(3, -4)}$

To plot the full graph:

• Midline: horizontal line $y=-4$
• Points repeat every 2 units along the x-axis, with peaks 2 units above the midline and troughs 2 units below.