Question

choose 4, the equation and 3 features

a \\(y = 2 \sin(x) + 6\\)

b \\(y = 2 \cos(x) + 6\\)

c \\(y = 2 \tan(x) + 6\\)

d the midline is 6

e the midline is 2

f the amplitude is 6

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Find the midline and amplitude

The maximum value of the curve is \(8\) (at \(x = 90^\circ\)) and the minimum value is \(4\) (at \(x = 270^\circ\)).

\[
\text{Midline} = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{8 + 4}{2} = 6
\]

\[
\text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{8 - 4}{2} = 2
\]

Step 2: Identify the trigonometric function and its equation

The graph passes through its midline at \(x = 0\) with a value of \(6\), rises to its maximum at \(x = 90^\circ\), returns to the midline at \(x = 180^\circ\), reaches its minimum at \(x = 270^\circ\), and returns to the midline at \(x = 360^\circ\). This matches the behavior of a sine wave:

\[
y = A \sin(x) + D
\]

Using amplitude \(A = 2\) and midline vertical shift \(D = 6\):

\[
y = 2 \sin(x) + 6
\]

Step 3: Select the correct options

Based on our findings:

• The correct equation is \(y = 2 \sin(x) + 6\) (Option A).
• The midline is \(6\) (Option D).
• The amplitude is \(2\) (Option G, which is implied by the equation and standard features, though not explicitly listed in the visible options A-F, we select the correct statements from the choices provided).

Looking at the choices provided in the image:

• A \(y = 2 \sin(x) + 6\) (Correct equation)
• D the midline is 6 (Correct feature)

(Note: Since the prompt asks to "Choose 4, the equation and 3 features", and only options A through F are fully visible, the remaining correct features from the complete set of options would be "the amplitude is 2" and "the period is 360".)

Answer:

The correct choices from the visible options are:

• A \(y = 2 \sin(x) + 6\)
• D the midline is 6