To compare the graph of [tex]\( g(x) = \frac{4}{5} x^2 \)[/tex] with the graph of [tex]\( f(x) = x^2 \)[/tex], consider the following steps:
1. Definition of Functions:
- [tex]\( f(x) \)[/tex] is defined as [tex]\( f(x) = x^2 \)[/tex].
- [tex]\( g(x) \)[/tex] is defined as [tex]\( g(x) = \frac{4}{5} x^2 \)[/tex].
2. Analysis of [tex]\( g(x) \)[/tex]:
- To determine how the graph of [tex]\( g(x) \)[/tex] compares to [tex]\( f(x) \)[/tex], examine the coefficient [tex]\(\frac{4}{5}\)[/tex] in [tex]\( g(x) \)[/tex].
- The coefficient [tex]\(\frac{4}{5}\)[/tex] is a positive number less than 1.
3. Vertical Transformation:
- When a function is multiplied by a coefficient [tex]\(k\)[/tex]:
- If [tex]\( 0 < k < 1 \)[/tex], the function experiences a vertical compression.
- If [tex]\( k > 1 \)[/tex], the function experiences a vertical stretch.
- If [tex]\( k < 0 \)[/tex], the function is flipped over the x-axis.
- For [tex]\( g(x) \)[/tex], [tex]\(k\)[/tex] is [tex]\(\frac{4}{5}\)[/tex], which is less than 1 but greater than 0. Therefore, [tex]\( g(x) \)[/tex] is a vertically compressed version of [tex]\( f(x) \)[/tex].
- The function is not flipped over the x-axis because the coefficient is positive.
4. Conclusion:
- The graph of [tex]\( g(x) = \frac{4}{5} x^2 \)[/tex] is the graph of [tex]\( f(x) = x^2 \)[/tex], but compressed vertically by a factor of [tex]\(\frac{4}{5}\)[/tex].
So, the correct comparison statement is:
B. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] compressed vertically.
