Question

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

\\(f(x) = \\) drop-down \\(\cos x\\) drop-down

drop-down options shown:
-2
2
-3
4
5

Explanation:

⚡ Using what you learned: Graphing Trigonometric Functions

Step 1: Find the midline and amplitude

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

The midline \( d \) is the average of the maximum and minimum values:

$$ d = \frac{-1 + (-5)}{2} = -3 $$

The amplitude \( A \) is the distance from the midline to a peak:

$$ A = \frac{-1 - (-5)}{2} = 2 $$

Step 2: Determine the function type and vertical reflection

At \( x = 0 \), the graph is at its maximum value, \( y = -1 \).
A standard cosine function, \( \cos(x) \), starts at its maximum value. Since the graph starts at a maximum at \( x = 0 \), there is no vertical reflection.

Thus, the coefficient in front of the cosine term (the amplitude) is positive:

$$ a = 2 $$

Step 3: Write the complete equation

Using the general form \( f(x) = a \cos(x) + d \):

$$ f(x) = 2\cos(x) - 3 $$

Comparing this to the dropdown options in the image:

• First dropdown (coefficient of \( \cos(x) \)): 2
• Second dropdown (trigonometric function): cosx
• Third dropdown (vertical shift): -3

Answer:

The values to select in the dropdowns to write the equation are:

• First box: 2
• Second box: cosx
• Third box: -3

The complete equation is:

$$ f(x) = 2\cos(x) - 3 $$