Answer :
To compare the graph of [tex]\( G(x) = \frac{4}{5} x^2 \)[/tex] with the graph of [tex]\( F(x) = x^2 \)[/tex], let's analyze how the function [tex]\( G(x) \)[/tex] transforms [tex]\( F(x) \)[/tex].
1. Understanding [tex]\(F(x)\)[/tex]:
- [tex]\( F(x) = x^2 \)[/tex]
- This is a standard parabola that opens upward with its vertex at the origin (0, 0).
2. Understanding [tex]\(G(x)\)[/tex]:
- [tex]\( G(x) = \frac{4}{5} x^2 \)[/tex]
- This transformation involves multiplying the function [tex]\( x^2 \)[/tex] by a coefficient, [tex]\(\frac{4}{5}\)[/tex].
3. Effect of the Coefficient [tex]\(\frac{4}{5}\)[/tex]:
- A coefficient less than 1 but greater than 0 in front of [tex]\( x^2 \)[/tex] compresses the graph vertically.
- This means that for any given [tex]\( x \)[/tex], the value of [tex]\( G(x) \)[/tex] will be [tex]\(\frac{4}{5}\)[/tex] times the value of [tex]\( F(x) \)[/tex], making [tex]\( G(x) \)[/tex] shorter compared to [tex]\( F(x) \)[/tex].
4. Comparison Statements:
- A. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] compressed vertically and flipped over the [tex]\( x \)[/tex]-axis.
- This is incorrect because there is no negative coefficient indicating a flip over the [tex]\( x \)[/tex]-axis.
- B. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] stretched vertically.
- This is incorrect because a coefficient less than 1 indicates a compression, not a stretch.
- C. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] stretched vertically and flipped over the [tex]\( x \)[/tex]-axis.
- This is incorrect for the same reasons as options A and B.
- D. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] compressed vertically.
- This is correct because a coefficient of [tex]\(\frac{4}{5}\)[/tex] in front of [tex]\( x^2 \)[/tex] ensures a vertical compression.
Thus, the best statement that compares the graphs is:
D. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] compressed vertically.
1. Understanding [tex]\(F(x)\)[/tex]:
- [tex]\( F(x) = x^2 \)[/tex]
- This is a standard parabola that opens upward with its vertex at the origin (0, 0).
2. Understanding [tex]\(G(x)\)[/tex]:
- [tex]\( G(x) = \frac{4}{5} x^2 \)[/tex]
- This transformation involves multiplying the function [tex]\( x^2 \)[/tex] by a coefficient, [tex]\(\frac{4}{5}\)[/tex].
3. Effect of the Coefficient [tex]\(\frac{4}{5}\)[/tex]:
- A coefficient less than 1 but greater than 0 in front of [tex]\( x^2 \)[/tex] compresses the graph vertically.
- This means that for any given [tex]\( x \)[/tex], the value of [tex]\( G(x) \)[/tex] will be [tex]\(\frac{4}{5}\)[/tex] times the value of [tex]\( F(x) \)[/tex], making [tex]\( G(x) \)[/tex] shorter compared to [tex]\( F(x) \)[/tex].
4. Comparison Statements:
- A. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] compressed vertically and flipped over the [tex]\( x \)[/tex]-axis.
- This is incorrect because there is no negative coefficient indicating a flip over the [tex]\( x \)[/tex]-axis.
- B. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] stretched vertically.
- This is incorrect because a coefficient less than 1 indicates a compression, not a stretch.
- C. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] stretched vertically and flipped over the [tex]\( x \)[/tex]-axis.
- This is incorrect for the same reasons as options A and B.
- D. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] compressed vertically.
- This is correct because a coefficient of [tex]\(\frac{4}{5}\)[/tex] in front of [tex]\( x^2 \)[/tex] ensures a vertical compression.
Thus, the best statement that compares the graphs is:
D. The graph of [tex]\( G(x) \)[/tex] is the graph of [tex]\( F(x) \)[/tex] compressed vertically.
