To determine the effect of the transformation of the graph of the function [tex]\( f(x) = x^2 \)[/tex] to [tex]\( h(x) = 3x^2 - 7 \)[/tex], let's analyze each component of the transformation separately.
1. Vertical Stretch:
- The original function is [tex]\( f(x) = x^2 \)[/tex].
- In [tex]\( h(x) = 3x^2 - 7 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is 3.
- When a function is multiplied by a constant factor greater than 1, the graph is vertically stretched.
- Thus, the graph of [tex]\( f(x) = x^2 \)[/tex] is vertically stretched by a factor of 3.
2. Vertical Shift:
- In [tex]\( h(x) = 3x^2 - 7 \)[/tex], there is a [tex]\( -7 \)[/tex] term.
- This term indicates a vertical shift.
- Specifically, subtracting 7 from the function means the entire graph is shifted downward by 7 units.
Considering both transformations together:
- The graph is vertically stretched by a factor of 3.
- The graph is shifted 7 units downward.
Combining these, the correct effect on the graph is described in option:
C. The graph of [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 3 and shifted 7 units down.
