Question

find an equation for a cosine function having an amplitude of 2, period of \\(\pi\\), horizontal shift of \\(2\pi\\) to the right, and a vertical shift of -3.

\\y = ? \cos( x + \pi) + \\

Explanation:

⚡ Using what you learned: Amplitude, Period, Phase Shift

Step 1: Identify the general form and parameters

The general equation for a transformed cosine function is:

$$ y = A \cos(B(x - C)) + D $$

which can be expanded to:

$$ y = A \cos(Bx - BC) + D $$

From the problem statement:

• Amplitude \( |A| = 2 \implies A = 2 \)
• Period \( = \pi \)
• Horizontal shift (phase shift) \( C = 2\pi \) (to the right)
• Vertical shift \( D = -3 \)

Step 2: Calculate the frequency coefficient B

The formula relating the period to \( B \) is:

$$ \text{Period} = \frac{2\pi}{B} $$

$$ \pi = \frac{2\pi}{B} \implies B = 2 $$

Step 3: Determine the horizontal phase term

Substitute \( B = 2 \) and \( C = 2\pi \) into the expression \( B(x - C) \):

$$ B(x - C) = 2(x - 2\pi) = 2x - 4\pi $$

This matches the template form \( [ \quad ]x + [ \quad ]\pi \):

$$ 2x + (-4)\pi $$

Step 4: Assemble the final equation

Substitute all parameters into the template \( y = [A] \cos([B]x + [C']\pi) + [D] \):

$$ y = 2 \cos(2x - 4\pi) - 3 $$

Answer:

$$ y = 2 \cos(2x - 4\pi) - 3 $$

The values for the boxes from left to right are:

• First box: 2
• Second box: 2
• Third box: -4
• Fourth box: -3