Phoebe claims that the equation for g ( x ) is g ( x ) = − ( x + 2 ) 2 because the following transformations were performed. The function was reflected over the x-axis. The function was shifted two units to the right. Phoebe's equation is incorrect. Describe Phoebe's mistake and write a correct equation for g ( x ) .

Phoebe's mistake in the equation for g(x) is that she incorrectly reflected the function over the x-axis and shifted it two units to the right. Let's break down the transformations and correct her equation:

1. Reflection over the x-axis: When a function is reflected over the x-axis, the sign of the function's y-values changes. In this case, the function should have been reflected downward, not upward. Phoebe's equation reflects the function upward, which is incorrect.

2. Shift two units to the right: To shift a function two units to the right, we replace x with (x - 2) in the equation. Phoebe's equation mistakenly shifts the function two units to the left by adding 2 instead of subtracting 2.

To correct Phoebe's equation, we need to apply the correct transformations:

1. Reflecting the function downward over the x-axis: We can achieve this by multiplying the function by -1. So, the corrected equation becomes: g(x) = -[(x + 2)^2].

2. Shifting the function two units to the right: We replace x with (x - 2) in the equation. So, the corrected equation becomes: g(x) = -[(x - 2 + 2)^2] = -[x^2].

Therefore, the correct equation for g(x) is g(x) = -x^2, which reflects the function downward over the x-axis and does not include a horizontal shift.