Question

12 choose the trig function that represents the graph below. also choose the 4 features that are true in the graph.

choose 4 the equation and correct features of the graph

a \\(y = -4 \cos(2x)\\)
b \\(y = -4 \sin(2x)\\)
c \\(y = -4 \tan(x)\\)
d midline is -4
e midline is 0
f amplitude is -4
g the amplitude is 4
h amplitude is 2
i midline is 2
j period is \\(\pi\\) or 180
k period is \\(2\pi\\) or 360

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Identify the midline and amplitude

The maximum value of the graph is \(4\) and the minimum value is \(-4\).

The midline is the horizontal line halfway between the maximum and minimum values:
\[y = \frac{4 + (-4)}{2} = 0\]

The amplitude is the distance from the midline to either the maximum or minimum value:
\[\text{Amplitude} = 4 - 0 = 4\]

(Note: Amplitude is always a positive distance, so it is \(4\), not \(-4\).)

Step 2: Identify the period

The graph starts a full cycle at \(x = 0\) (at its minimum value of \(-4\)) and completes one full cycle when it returns to its minimum value at \(x = 180^\circ\) (or \(\pi\) radians).

Therefore, the period is:
\[\text{Period} = 180^\circ \text{ (or } \pi\text{)}\]

Step 3: Determine the equation

Since the graph starts at its minimum value on the \(y\)-axis (\(x = 0\)), it represents a reflected cosine function of the form:
\[y = -A \cos(Bx)\]

Using the amplitude \(A = 4\):
\[y = -4 \cos(Bx)\]

The formula for the period in radians is:
\[\text{Period} = \frac{2\pi}{B}\]

Substitute the period \(\pi\):
\[\pi = \frac{2\pi}{B} \implies B = 2\]

Thus, the equation is:
\[y = -4 \cos(2x)\]

Answer:

The 4 correct choices are:

• A \(y = -4 \cos(2x)\)
• E midline is 0
• G the amplitude is 4
• J period is \(\pi\) or 180