Question

question 7 (1 point)
when the value of \\(k\\) is between 0 and 1, the function \\(g(x) = (kx)^2\\) has what relationship to the base function \\(f(x) = x^2\\)?

a) \\(f(x)\\) is reflected and stretched vertically

b) \\(f(x)\\) is stretched vertically

c) \\(f(x)\\) is stretched horizontally

d) \\(f(x)\\) is reflected and stretched horizontally

Explanation:

Analyze the transformation type

The given function is \(g(x) = (kx)^2\) and the base function is \(f(x) = x^2\).
Since the constant \(k\) is multiplied directly to the input variable \(x\) inside the function argument, this represents a horizontal transformation. Specifically, a transformation of the form \(f(kx)\) represents a horizontal stretch or compression.

Determine the effect of the parameter value

We are given that the value of \(k\) is between 0 and 1, which means \(0 < k < 1\).
Using the Horizontal Stretching and Function Transformations knowledge points:

• When \(0 < k < 1\), the graph of \(y = f(kx)\) is stretched horizontally by a factor of \(\frac{1}{k}\).
• Since \(k > 0\), there is no reflection across the y-axis (which would require \(k < 0\)).

Match with the given options

Let's evaluate the choices:

• A) \(f(x)\) is reflected and stretched vertically (Incorrect: no reflection, and the transformation is horizontal).
• B) \(f(x)\) is stretched vertically (Incorrect: the transformation is horizontal).
• C) \(f(x)\) is stretched horizontally (Correct: since \(0 < k < 1\), it is a horizontal stretch).
• D) \(f(x)\) is reflected and stretched horizontally (Incorrect: there is no reflection).

Answer:

• A) \(f(x)\) is reflected and stretched vertically
• B) \(f(x)\) is stretched vertically
• C) \(f(x)\) is stretched horizontally (Correct answer)
• D) \(f(x)\) is reflected and stretched horizontally