Question

1 matching 1 point
match each given definition with the correct word it defines.
the height of the graph from the midline
the length of a cycle
2 multiple choice 1 point
on the function $y = a \sin(b\theta)$, what is the following called: $|a|$?
\bigcirc frequency
\bigcirc midline
\bigcirc period
\bigcirc amplitude
options for matching: midline, amplitude, period, horizontal phase shift

Explanation:

Sub - Question 1 (Matching)

• For the definition "The height of the graph from the midline", in the context of trigonometric graphs (or periodic graphs), this is the definition of Amplitude.
• For the definition "The length of a cycle", in the context of periodic functions, this is the definition of Period.
• Midline is the horizontal line that the graph oscillates around, and Horizontal Phase Shift is the horizontal shift of the graph, so they don't match these definitions.

In the function \(y = a\sin(b\theta)\), the absolute value of \(a\), \(|a|\), represents the amplitude. The amplitude is the height of the graph from the midline. Frequency is related to \(b\) (frequency \(=\frac{b}{2\pi}\) for sine functions), the midline is a horizontal line (not related to \(|a|\)), and the period is \(\frac{2\pi}{|b|}\), so \(|a|\) is the amplitude.

Answer:

• "The height of the graph from the midline" - Amplitude
• "The length of a cycle" - Period

Sub - Question 2 (Multiple Choice)