From tech troubles to travel tips, IDNLearn.com has answers to all your questions. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
To find the graph of the function [tex]\( g(x) = f(4x) \)[/tex] where [tex]\( f(x) = x^2 \)[/tex], let's go through the process step by step:
1. Understand the function [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = x^2 \)[/tex] is a standard quadratic function. Its graph is a parabola that opens upwards with its vertex at the origin (0,0).
2. Substitute [tex]\( 4x \)[/tex] into [tex]\( f(x) \)[/tex]:
- [tex]\( g(x) = f(4x) \)[/tex].
- Since [tex]\( f(x) = x^2 \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\( 4x \)[/tex] in the function [tex]\( f(x) \)[/tex]. So, [tex]\( f(4x) = (4x)^2 \)[/tex].
3. Simplify [tex]\( g(x) \)[/tex]:
- When we simplify [tex]\( (4x)^2 \)[/tex]:
[tex]\[ g(x) = (4x)^2 = 16x^2 \][/tex]
Therefore, the function [tex]\( g(x) = 16x^2 \)[/tex].
4. Analyze the Transformation:
- The original function [tex]\( f(x) = x^2 \)[/tex] has been transformed into [tex]\( g(x) = 16x^2 \)[/tex].
- This transformation involves a horizontal compression by a factor of 4. To see why, note that [tex]\( x \)[/tex] in the original function [tex]\( f \)[/tex] is replaced by [tex]\( 4x \)[/tex]. In terms of the graph, every point [tex]\( (x, y) \)[/tex] on the graph of [tex]\( f \)[/tex] will be mapped to [tex]\( (\frac{x}{4}, y) \)[/tex] on the graph of [tex]\( g \)[/tex].
- Another way to see this is that [tex]\( f(4x) \)[/tex] will produce the same output as [tex]\( f(x) \)[/tex] but for an input that is [tex]\( \frac{1}{4} \)[/tex] of the original [tex]\( x \)[/tex].
5. Sketch the Graph:
- The graph of [tex]\( g(x) = 16x^2 \)[/tex] will still be a parabola opening upwards.
- However, compared to [tex]\( f(x) \)[/tex], it will be much narrower (compressed horizontally) due to the factor of 16.
So, summarizing:
- The original graph [tex]\( f(x) = x^2 \)[/tex] is transformed into [tex]\( g(x) = 16x^2 \)[/tex].
- The new graph will be a vertically stretched and horizontally compressed parabola compared to [tex]\( f(x) = x^2 \)[/tex].
Therefore, the graph of [tex]\( g(x) \)[/tex] is a parabola which is narrower than the graph of [tex]\( f(x) = x^2 \)[/tex] and has the equation [tex]\( y = 16x^2 \)[/tex].
1. Understand the function [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = x^2 \)[/tex] is a standard quadratic function. Its graph is a parabola that opens upwards with its vertex at the origin (0,0).
2. Substitute [tex]\( 4x \)[/tex] into [tex]\( f(x) \)[/tex]:
- [tex]\( g(x) = f(4x) \)[/tex].
- Since [tex]\( f(x) = x^2 \)[/tex], we replace [tex]\( x \)[/tex] with [tex]\( 4x \)[/tex] in the function [tex]\( f(x) \)[/tex]. So, [tex]\( f(4x) = (4x)^2 \)[/tex].
3. Simplify [tex]\( g(x) \)[/tex]:
- When we simplify [tex]\( (4x)^2 \)[/tex]:
[tex]\[ g(x) = (4x)^2 = 16x^2 \][/tex]
Therefore, the function [tex]\( g(x) = 16x^2 \)[/tex].
4. Analyze the Transformation:
- The original function [tex]\( f(x) = x^2 \)[/tex] has been transformed into [tex]\( g(x) = 16x^2 \)[/tex].
- This transformation involves a horizontal compression by a factor of 4. To see why, note that [tex]\( x \)[/tex] in the original function [tex]\( f \)[/tex] is replaced by [tex]\( 4x \)[/tex]. In terms of the graph, every point [tex]\( (x, y) \)[/tex] on the graph of [tex]\( f \)[/tex] will be mapped to [tex]\( (\frac{x}{4}, y) \)[/tex] on the graph of [tex]\( g \)[/tex].
- Another way to see this is that [tex]\( f(4x) \)[/tex] will produce the same output as [tex]\( f(x) \)[/tex] but for an input that is [tex]\( \frac{1}{4} \)[/tex] of the original [tex]\( x \)[/tex].
5. Sketch the Graph:
- The graph of [tex]\( g(x) = 16x^2 \)[/tex] will still be a parabola opening upwards.
- However, compared to [tex]\( f(x) \)[/tex], it will be much narrower (compressed horizontally) due to the factor of 16.
So, summarizing:
- The original graph [tex]\( f(x) = x^2 \)[/tex] is transformed into [tex]\( g(x) = 16x^2 \)[/tex].
- The new graph will be a vertically stretched and horizontally compressed parabola compared to [tex]\( f(x) = x^2 \)[/tex].
Therefore, the graph of [tex]\( g(x) \)[/tex] is a parabola which is narrower than the graph of [tex]\( f(x) = x^2 \)[/tex] and has the equation [tex]\( y = 16x^2 \)[/tex].
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.
