To determine the effect on the graph of [tex]\( f(x) = x^2 \)[/tex] when it is transformed to [tex]\( h(x) = 5x^2 + 10 \)[/tex], let's analyze the transformation step-by-step.
1. Vertical Stretch by a Factor of 5:
- The function [tex]\( h(x) \)[/tex] is given by [tex]\( 5x^2 \)[/tex].
- Comparing this to [tex]\( f(x) = x^2 \)[/tex], you can see that each output [tex]\( x^2 \)[/tex] of the original function is multiplied by 5.
- This means every point on the original graph (e.g., if [tex]\( x = 1 \)[/tex], [tex]\( f(1) = 1^2 = 1 \)[/tex] and [tex]\( h(1) = 5 \cdot 1^2 = 5 \)[/tex]) is pulled further away from the x-axis by a factor of 5.
- Therefore, this is a vertical stretch by a factor of 5.
2. Vertical Shift Upwards by 10 Units:
- After applying the vertical stretch, the entire graph is then shifted upwards by 10 units.
- This upward shift is represented by the constant term [tex]\( +10 \)[/tex] in [tex]\( h(x) = 5x^2 + 10 \)[/tex].
- Essentially, each value of [tex]\( 5x^2 \)[/tex] is increased by 10.
Combining these transformations, we can see that:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is first vertically stretched by a factor of 5.
- The entire graph is then shifted 10 units upwards.
So, the correct answer is:
A. The graph of [tex]\( f(x) \)[/tex] is vertically stretched by a factor of 5 and shifted 10 units up.
