Question

what is the effect on the graph of $f(x) = x^2$ when it is transformed to $h(x) = 3x^2 - 7$?
a. the graph of $f(x)$ is vertically stretched by a factor of 3 and shifted 7 units to the right.
b. the graph of $f(x)$ is horizontally compressed by a factor of 3 and shifted 7 units to the right.
c. the graph of $f(x)$ is vertically stretched by a factor of 3 and shifted 7 units down.
d. the graph of $f(x)$ is horizontally stretched by a factor of 3 and shifted 7 units down.

Explanation:

Step1: Identify vertical stretch factor

For $f(x)=x^2$ transformed to $h(x)=3x^2-7$, the coefficient 3 multiplying $x^2$ means a vertical stretch by factor 3.

Step2: Identify vertical shift

The $-7$ at the end of $h(x)=3x^2-7$ means a shift 7 units down.

Step3: Match to options

Eliminate options with horizontal transformations (A, B, D) as there is no horizontal change to $x$ itself.

Answer:

C. The graph of f(x) is vertically stretched by a factor of 3 and shifted 7 units down.