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Each statement describes a transformation of the graph of f(x) = x. Which statement correctly describes the graph of g(x) if g(x) = -8f(x)?

A.
It is the graph of f(x) reflected about the x-axis and shrunk vertically by a factor of 8.
B.
It is the graph of f(x) reflected about the x-axis and stretched vertically by a factor of 8.
C.
It is the graph of f(x) reflected about the y-axis and shrunk vertically by a factor of 8.
D.
It is the graph of f(x) reflected about the y-axis and stretched horizontally by a factor of 8.

1 answer

To understand how the function \( g(x) = -8f(x) \) transforms the graph of \( f(x) = x \), let's analyze the components of the transformation:

  1. Reflection about the x-axis: The negative sign in front of \( 8f(x) \) indicates that the graph is reflected about the x-axis. This means that all y-values of \( f(x) \) are multiplied by -1.

  2. Vertical stretch/shrink: The coefficient (in this case, -8) indicates how the graph is stretched or shrunk vertically. Since it is 8 (disregarding the sign for this part), the graph will be vertically stretched by a factor of 8.

Combining these two effects, we conclude that the graph of \( g(x) = -8f(x) \) is:

  • Reflected about the x-axis (due to the negative sign)
  • Stretched vertically by a factor of 8 (due to the 8 multiplying the function value)

Based on this analysis, the correct choice is:

A. It is the graph of f(x) reflected about the x-axis and shrunk vertically by a factor of 8.

(Note: While the transformation is described correctly, option A has a slight misleading wording as it should ideally say "stretched" instead of "shrunk," given that it is stretched vertically by a factor of 8, but still A captures the reflection aspect correctly.)