To understand the effect on the graph of [tex]\( f(x) = x^2 \)[/tex] when it is transformed to [tex]\( h(x) = \frac{1}{5} x^2 + 12 \)[/tex], let's break down the transformations step by step.
1. Vertical Compression:
- The term [tex]\( \frac{1}{5} x^2 \)[/tex] indicates a vertical compression.
- A vertical compression occurs when the y-values of the graph are reduced.
- Since the new equation has [tex]\( \frac{1}{5} \)[/tex] as a coefficient of [tex]\( x^2 \)[/tex], the graph is vertically compressed by a factor of 5.
2. Vertical Shift:
- The term [tex]\( +12 \)[/tex] in [tex]\( h(x) = \frac{1}{5} x^2 + 12 \)[/tex] indicates a vertical shift.
- Adding 12 to the function shifts the entire graph up by 12 units.
Now, let’s summarize these transformations:
- The graph is vertically compressed by a factor of 5 due to the [tex]\( \frac{1}{5} \)[/tex] multiplier.
- The graph is shifted vertically upwards by 12 units due to the [tex]\( +12 \)[/tex].
Therefore, the correct description of the transformation is:
C. The graph of [tex]\( f(x) \)[/tex] is vertically compressed by a factor of 5 and shifted 12 units up.
