AP PreCalculus Practice Quiz: Transformations of Functions

Correct Answer: B

According to the content, the function g(x) = f(x) + k is an additive transformation that results in a vertical translation by k units. In this case, k = -7, which corresponds to a vertical translation of the graph of f by 7 units down.

Correct Answer: B

The content states that g(x) = f(x+h) results in a horizontal translation of the graph of f by -h units. Here, h = 3, so the translation is by -3 units, which is 3 units to the left.

Correct Answer: C

The content defines a multiplicative transformation as g(x) = a f(x) or g(x) = f(bx). The function g(x) = 4f(x) fits the form g(x) = a f(x) with a=4. The other options are all additive transformations.

Correct Answer: B

The function g(x) = a f(x) results in a vertical dilation of the graph of f by a factor of |a|. In this case, a = 1/2, so the graph is vertically dilated (compressed) by a factor of 1/2.

Correct Answer: C

The content states that g(x) = f(bx) results in a horizontal dilation of the graph of f by a factor of |1/b|. Here, b = 3, so the horizontal dilation is by a factor of 1/3.

Correct Answer: B

A translation 2 units to the right corresponds to f(x+h) with -h = 2, so h = -2, giving f(x-2). A translation 5 units down corresponds to f(x)+k with k = -5. Combining these gives g(x) = f(x-2) - 5.

Correct Answer: A

This is a combination of a multiplicative transformation and an additive transformation. The g(x) = a f(x) part is a = -4, which is a vertical dilation by |a|=4 (and a reflection). The g(x) = f(x) + k part is k = 9, a vertical translation up by 9 units. The dilation is applied first.

Correct Answer: C

The transformation g(x) = f(bx) with b=2 results in a horizontal dilation by a factor of 1/|b| = 1/2. This transformation affects the domain of the function. To find the new domain, we multiply the endpoints of the original domain by the dilation factor: [-10 * (1/2), 10 * (1/2)], which results in [-5, 5].

Correct Answer: B

The transformation g(x) = f(x) + k is a vertical translation by k units. This transformation affects the range of the function by shifting it. Here, k=5, so the new range is found by adding 5 to the endpoints of the original range: [-4 + 5, 8 + 5], which results in [1, 13].

Correct Answer: C

A horizontal dilation by a factor of 4 corresponds to g(x) = f(bx) where the factor is |1/b|. So, |1/b| = 4, which means b = 1/4. A horizontal translation 3 units to the left corresponds to g(x) = f(x+h) where -h = -3, so h = 3. Combining these gives g(x) = f(b(x+h)) = f(1/4 * (x+3)).

Correct Answer: B

The transformation is of the form g(x) = f(bx). In this case, f(x/5) can be written as f((1/5)x), so b = 1/5. The horizontal dilation is by a factor of |1/b|. Therefore, the factor is |1/(1/5)| = 5.

Correct Answer: A

The range of f(x) is [10, 20]. First, apply the multiplicative transformation a=-2. This multiplies the range by -2, giving [-40, -20] and reversing the order to [-40, -20]. Second, apply the additive transformation k=-3. This subtracts 3 from the endpoints, giving [-40 - 3, -20 - 3], which results in the new range [-43, -23].

Correct Answer: A

To identify the horizontal transformations, we must rewrite the input in the form b(x+h). The input is 6-2x, which can be factored as -2(x-3). So, g(x) = 5f(-2(x-3)) + 1. Here, b=-2 and h=-3. The horizontal dilation is by a factor of |1/b| = |1/-2| = 1/2. The horizontal translation is by -h = -(-3) = 3 units, which is to the right.

Correct Answer: B

The provided content explicitly states that g(x) = f(x) + k and g(x) = f(x+h) are additive transformations of the function f. They involve adding or subtracting a constant either to the function's output or its input.