Question

which of the following describes the graph of $-3 \cdot f(x)$?

a. the graph of $f(x)$ is widened horizontally and reflected about the $y$-axis.

b. the graph of $f(x)$ is widened horizontally and reflected about the $x$-axis.

c. the graph of $f(x)$ is stretched vertically and reflected about the $x$-axis.

d. the graph of $f(x)$ is stretched vertically and reflected about the $y$-axis.

Explanation:

Step1: Recall Transformations

For a function \( y = a \cdot f(x) \), the coefficient \( a \) affects vertical transformations. If \( |a|>1 \), it's a vertical stretch; if \( 0<|a|<1 \), vertical compression. A negative \( a \) reflects over the \( x \)-axis.

Step2: Analyze \( -3 \cdot f(x) \)

Here, \( a = -3 \). \( | - 3|=3>1 \), so vertical stretch. The negative sign means reflection over the \( x \)-axis. Horizontal transformations involve changes to \( x \) (like \( f(bx) \)), not the coefficient of \( f(x) \). So options about horizontal widening (A, B) are wrong. Reflection is over \( x \)-axis, not \( y \)-axis (D is wrong). So C is correct.

Answer:

C. The graph of \( f(x) \) is stretched vertically and reflected about the \( x \)-axis.