Question

question 2 (1 point) saved
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 stretched vertically

b) \\(f(x)\\) is reflected and 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 \(g(x) = (kx)^2\) where the base function is \(f(x) = x^2\). This represents a transformation of the form \(g(x) = f(kx)\). Since the multiplication by the constant \(k\) occurs inside the function argument before squaring, this is a horizontal transformation.

Determine the effect of the parameter

For a horizontal transformation \(f(kx)\):

• If \(|k| > 1\), the graph is horizontally compressed by a factor of \(\frac{1}{|k|}\).
• If \(0 < |k| < 1\), the graph is horizontally stretched by a factor of \(\frac{1}{|k|}\).
• Since \(k\) is between \(0\) and \(1\) (\(0 < k < 1\)), the graph of \(f(x)\) is stretched horizontally.

Check for reflections

Since \(k\) is positive (\(0 < k < 1\)), there is no negative sign inside or outside the function. Therefore, there is no reflection across either axis.

Match with the options

• Option A suggests a vertical stretch, which is incorrect because the transformation is horizontal.
• Option B suggests a reflection and vertical stretch, which is incorrect.
• Option C suggests a horizontal stretch, which matches our finding.
• Option D suggests a reflection and horizontal stretch, which is incorrect because there is no reflection.

Answer:

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