Period and amplitude of a sinusoid from its graph

Period is the horizontal length of one full cycle. Measure from one peak to the next: the graph reaches its maximum of 3 at $x=3$ and again at $x=11$, so the period is $11-3=8$. The minima agree: $y=-3$ at $x=7$ and $x=15$, also 8 apart.

Amplitude is the distance from the midline to a peak. The maximum is 3 and the minimum is $-3$, so the midline is $$\frac{3+(-3)}{2}=0$$ and the amplitude is $3-0=3$. Equivalently, half the distance from minimum to maximum: $\frac{3-(-3)}{2}=3$.

The graph has a period of 8 and an amplitude of 3, which is choice B.

Each choice

• A. A period of 4 is the distance between neighboring zeros, such as $x=1$ and $x=5$. That is only half a cycle; a full cycle returns to the same height going the same direction.
• B (correct). Peaks at 3 and 11 give a period of 8, and the graph rises 3 above its midline of 0.
• C. Both numbers are off by a factor of 2, in opposite directions. Zero to zero is half a period, and 6 is the full height from minimum to maximum, which is twice the amplitude.
• D. The period matches, but 6 is the distance from the lowest point to the highest. Amplitude is measured from the midline, so it is half of that.