Question

choose the trig function that represents the graph below. also choose the 4 features

choose 4 the equation and correct features of the graph

a period is \\(2\pi\\) or 360
b \\(y = 3 \sin(4x)\\)
c amplitude is 3
d midline is 3
e amplitude is 4
f midline is 2
g amplitude is 4
h period is \\(\pi\\) or 180
i period is \\(\pi/2\\) or 90
j \\(y = \tan(x)\\)
k \\(y = 3 \cos(4x) + 2\\)

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Find the midline and amplitude

The maximum value of the graph is \(5\) and the minimum value is \(-1\).

The midline is the horizontal line halfway between the maximum and minimum values:

$$ \text{midline} = \frac{\text{maximum} + \text{minimum}}{2} = \frac{5 + (-1)}{2} = 2 $$

The amplitude is the vertical distance from the midline to either the maximum or minimum value:

$$ \text{amplitude} = \text{maximum} - \text{midline} = 5 - 2 = 3 $$

Step 2: Find the period

The graph starts at its maximum value of \(5\) at \(x = 0\), which indicates a cosine function with no horizontal phase shift.

The first full cycle ends at \(x = \frac{\pi}{2}\) (or \(90^\circ\)), where the graph reaches its next maximum. Thus, the period is:

$$ \text{period} = \frac{\pi}{2} \text{ or } 90^\circ $$

Step 3: Determine the equation

Using the standard form \(y = A \cos(Bx) + D\):

• Amplitude \(A = 3\)
• Midline vertical shift \(D = 2\)
• Frequency \(B = \frac{2\pi}{\text{period}} = \frac{2\pi}{\pi/2} = 4\)

This gives the equation:

$$ y = 3 \cos(4x) + 2 $$

Answer:

The 4 correct features and equation are:

• C amplitude is 3
• F midline is 2
• I period is \(\pi/2\) or 90
• K \(y = 3 \cos(4x) + 2\)