Question

Suppose the function f(x) = x^2 is transformed and the new rule for the translated function, g(x), is g(x) = (3x)^2. Describe this transformation.

Answer 1

The transformation from f(x) = x^2 to g(x) = (3x)^2 is a horizontal compression by a factor of 1/3. For any point on the graph of f(x), the corresponding point on g(x) will have its x-coordinate compressed by 1/3 and the y-coordinate scaled by a factor of 9, as the entire function is squared.

Step-by-step explanation:

When we examine the transformation from f(x) = x^2 to g(x) = (3x)^2, we are looking at a transformation of the original function involving a horizontal scaling. The square function, which is f(x) = x^2, gets affected by the transformation to g(x), specifically a horizontal compression since the x-coordinate is multiplied by a factor before it is squared. In the case of g(x) = (3x)^2, every x value from the original function, f(x), is multiplied by 3 before the squaring process.

To better understand this transformation, consider how a point (a, b) where b = a^2 on the graph of f(x) would change on g(x). With the transformation to g(x), the point would become ((1/3)a, 9b) since (3*(1/3)a)^2 = 9a^2. This shows that the graph of g(x) is a horizontal compression of the graph of f(x) by a factor of 1/3, which means the horizontal distances on the graph of g(x) are one third of those on the graph of f(x), while the vertical dimensions are multiplied by 9, as the entire function is squared.

Answer 2

The function g(x) = (3x)^2 is a transformation of the original function f(x) = x^2. In this case, g(x) is obtained by applying a horizontal stretch with a scale factor of 1/3 to the function f(x).

When you have g(x) = (3x)^2, it means that every x-value in the original function f(x) is multiplied by 3 to obtain the transformed function g(x). This is a horizontal stretch by a factor of 1/3.

The general form of the transformation of a function f(x) to obtain g(x) is g(x) = f(kx), where k is a constant. In this case, k = 3, indicating the horizontal stretch.

So, the transformation from f(x) = x^2 to g(x) = (3x)^2 is a horizontal stretch by a factor of 1/3.