Question

Describe the transformation of f(x)=x² represented by g(x)=-3(x²)²-1?

Answer 1

The transformation of f(x)=x² by g(x)=-3(x²)²-1 includes squaring the function, a vertical stretch by a factor of 3 combined with a reflection across the x-axis, and a vertical translation downward by 1 unit.

Step-by-step explanation:

The transformation of the function f(x)=x² represented by g(x)=-3(x²)²-1 involves several steps. First, the f(x) is squared, indicated by the (x²)² term. This represents a power-raising transformation, which drastically changes the graph. After raising the power, the entire function is multiplied by -3, which is a vertical stretch by a factor of 3 and a reflection across the x-axis. Lastly, -1 is subtracted from the function which translates it downward by 1 unit.

To summarize the transformations in order:

1. Squaring the function: This turns f(x)=x² into f(x)=x⁴ (not shown explicitly in g(x), but implied by the squaring of x²).
2. Vertical stretch and reflection: Multiplying by -3 stretches the function vertically by a factor of 3 and reflects it across the x-axis.
3. Vertical translation: Subtracting 1 translates the function down by 1 unit.