Question

18 choose the correct trig function that represents the graph below and the features it represents.
choose 4, the equation and 3 features
a \\(y = 3 \tan(x) + 2\\)
b the midline is 3
c the midline is 2
d the period is \\(\pi\\) 180
e the amplitude is 2
f \\(y = 3 \cos(x) + 2\\)
g the amplitude is 3
h \\(y = 3 \sin(x) + 2\\)
i the period is \\(2\pi\\) 360

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Identify the midline

The midline is the horizontal line halfway between the maximum and minimum values of the graph.

$$ \text{Maximum value} = 5 $$

$$ \text{Minimum value} = -1 $$

$$ \text{Midline} = \frac{5 + (-1)}{2} = 2 $$

Thus, the midline is \( y = 2 \).

Step 2: Identify the amplitude

The amplitude is the vertical distance between the midline and a peak (or trough).

$$ \text{Amplitude} = \text{Maximum} - \text{Midline} = 5 - 2 = 3 $$

Step 3: Identify the period

The period is the horizontal distance required for the function to complete one full cycle.
Looking at the graph, a peak occurs at \( x = 2 \) and the next peak occurs at \( x \approx 8.28 \) (which is \( 2 + 2\pi \)).
Alternatively, the graph starts its rising phase through the midline \( y = 2 \) at \( x = 0 \) and completes one full wave cycle when it reaches the midline going upward again at \( x \approx 6.28 \) (which is \( 2\pi \)).

$$ \text{Period} = 2\pi \text{ (or } 360^\circ\text{)} $$

Step 4: Determine the equation

Since the graph passes through its midline \( y = 2 \) at \( x = 0 \) and goes upward, it represents a sine function with no horizontal phase shift:

$$ y = A \sin(x) + D $$

Substituting the amplitude \( A = 3 \) and midline vertical shift \( D = 2 \):

$$ y = 3 \sin(x) + 2 $$

Answer:

The 4 correct choices are:

• C the midline is 2
• G the amplitude is 3
• H \( y = 3 \sin(x) + 2 \)
• I the period is \( 2\pi \text{ } 360 \)