Question

Identify the a, h, and k of the function g(x) = 2 cos(2x) + 1.

a) a = 2, h = 2, k = 1
b) a = 2, h = 0, k = 1
c) a = 1, h = 2, k = 2
d) a = 1, h = 0, k = 2

Answer 1

The function g(x) = 2 cos(2x) + 1 has an amplitude (a) of 2, no horizontal shift (h), and a vertical shift (k) of 1, making the correct choice option (b), a = 2, h = 0, k = 1.

Step-by-step explanation:

To identify the parameters a, h, and k for the function g(x) = 2 cos(2x) + 1, we look at the standard form of a cosine function, which is f(x) = a cos(b(x - h)) + k, where:

• a is the amplitude of the wave,
• b is the coefficient that affects the period of the wave,
• h is the horizontal shift, also known as the phase shift,
• k is the vertical shift.

Comparing this with the given function, we can see that:

• The amplitude a is 2, as it is the coefficient in front of the cosine function.
• The coefficient b is also 2, but since this affects the period and not the phase shift, it does not correspond to h in our standard form.
• There is no (x - h) term present, so h is 0, indicating that there is no horizontal shift.
• The vertical shift k is +1, visible as the constant added to the cosine.

Therefore, the correct identification of parameters for g(x) = 2 cos(2x) + 1 is a = 2, h = 0, and k = 1, which makes option (b) the correct choice.