Question

15 choose the drop down value that will write the equation of the given graph below.

\\(f(x) = \\)

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Identify the midline and amplitude

The maximum value of the function is \( y = -1 \) (at \( x = 90^\circ \) and \( x = -270^\circ \)).
The minimum value of the function is \( y = -9 \) (at \( x = 270^\circ \) and \( x = -90^\circ \)).

The midline \( d \) is the average of the maximum and minimum values:
\[ d = \frac{-1 + (-9)}{2} = -5 \]

The amplitude \( a \) is the distance from the midline to a peak:
\[ a = \frac{-1 - (-9)}{2} = 4 \]

Step 2: Determine the trigonometric function and phase shift

At \( x = 0 \), the graph passes through its midline at \( y = -5 \) and goes upwards. This is the characteristic behavior of an unshifted sine function:
\[ f(x) = a \sin(bx) + d \]

Since it starts at the midline and goes up, the coefficient of the sine term is positive:
\[ a = 4 \]

Step 3: Find the period and frequency coefficient

The graph completes one full cycle from \( x = 0^\circ \) to \( x = 360^\circ \).
Therefore, the period \( T \) is \( 360^\circ \).

Using degrees, the frequency coefficient \( b \) is:
\[ b = \frac{360^\circ}{\text{Period}} = \frac{360^\circ}{360^\circ} = 1 \]

Thus, the equation is:
\[ f(x) = 4\sin(x) - 5 \]

Answer:

The drop-down values to write the equation are:

• First box: \( 4 \)
• Second box: \( \sin(x) \) (or \( \sin(1x) \) depending on options)
• Third box: \( - 5 \)