Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.