Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
To compare the graphs of the functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = (2x)^2 \)[/tex], we need to analyze the effect of the transformation on the graph.
1. Identify the original function and the transformed function:
- The original function is [tex]\( f(x) = x^2 \)[/tex].
- The transformed function is [tex]\( g(x) = (2x)^2 \)[/tex].
2. Simplify the transformed function:
[tex]\[ g(x) = (2x)^2 = 4x^2 \][/tex]
3. Analyze the transformation:
- The relationship [tex]\( g(x) = 4x^2 \)[/tex] indicates that for each value of [tex]\( x \)[/tex], the output value of [tex]\( g(x) \)[/tex] is 4 times the corresponding output value of [tex]\( f(x) \)[/tex].
- This transformation has two components to consider:
- The coefficient 4 in front of [tex]\( x^2 \)[/tex], which indicates a vertical stretch.
- The term [tex]\( 2x \)[/tex] inside the parentheses, which affects the [tex]\( x \)[/tex]-coordinate.
4. Consider the impact of the term [tex]\( 2x \)[/tex]:
- The term [tex]\( 2x \)[/tex] means that every [tex]\( x \)[/tex]-value in the function [tex]\( f(x) \)[/tex] is multiplied by 2 in the function [tex]\( g(x) \)[/tex]. This does not affect the overall shape of the parabola but changes the location of points on the graph.
- Specifically, a factor of [tex]\( 2 \)[/tex] inside the function argument [tex]\( (2x) \)[/tex] corresponds to a horizontal compression. This is because multiplying [tex]\( x \)[/tex] by a factor of 2 actually squeezes the graph towards the y-axis.
5. Determine the correct comparison:
- Horizontally stretching means making the graph wider, effectively multiplying [tex]\( x \)[/tex]-values by a factor greater than 1.
- Horizontally compressing means making the graph narrower, effectively multiplying [tex]\( x \)[/tex]-values by a factor between 0 and 1 (the reciprocal of 2 in this case, which is [tex]\( \frac{1}{2} \)[/tex]).
- Vertically stretching would involve multiplying [tex]\( y \)[/tex]-values by a factor greater than 1, but it does not directly pertain to horizontal compression.
- Shifting by 2 units to the right would involve adding or subtracting a constant from [tex]\( x \)[/tex], which is not the case here.
Since [tex]\( g(x) = (2x)^2 = 4x^2 \)[/tex] results from a horizontal compression of [tex]\( f(x) = x^2 \)[/tex] by a factor of 2, the correct statement is:
C. The graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 2.
1. Identify the original function and the transformed function:
- The original function is [tex]\( f(x) = x^2 \)[/tex].
- The transformed function is [tex]\( g(x) = (2x)^2 \)[/tex].
2. Simplify the transformed function:
[tex]\[ g(x) = (2x)^2 = 4x^2 \][/tex]
3. Analyze the transformation:
- The relationship [tex]\( g(x) = 4x^2 \)[/tex] indicates that for each value of [tex]\( x \)[/tex], the output value of [tex]\( g(x) \)[/tex] is 4 times the corresponding output value of [tex]\( f(x) \)[/tex].
- This transformation has two components to consider:
- The coefficient 4 in front of [tex]\( x^2 \)[/tex], which indicates a vertical stretch.
- The term [tex]\( 2x \)[/tex] inside the parentheses, which affects the [tex]\( x \)[/tex]-coordinate.
4. Consider the impact of the term [tex]\( 2x \)[/tex]:
- The term [tex]\( 2x \)[/tex] means that every [tex]\( x \)[/tex]-value in the function [tex]\( f(x) \)[/tex] is multiplied by 2 in the function [tex]\( g(x) \)[/tex]. This does not affect the overall shape of the parabola but changes the location of points on the graph.
- Specifically, a factor of [tex]\( 2 \)[/tex] inside the function argument [tex]\( (2x) \)[/tex] corresponds to a horizontal compression. This is because multiplying [tex]\( x \)[/tex] by a factor of 2 actually squeezes the graph towards the y-axis.
5. Determine the correct comparison:
- Horizontally stretching means making the graph wider, effectively multiplying [tex]\( x \)[/tex]-values by a factor greater than 1.
- Horizontally compressing means making the graph narrower, effectively multiplying [tex]\( x \)[/tex]-values by a factor between 0 and 1 (the reciprocal of 2 in this case, which is [tex]\( \frac{1}{2} \)[/tex]).
- Vertically stretching would involve multiplying [tex]\( y \)[/tex]-values by a factor greater than 1, but it does not directly pertain to horizontal compression.
- Shifting by 2 units to the right would involve adding or subtracting a constant from [tex]\( x \)[/tex], which is not the case here.
Since [tex]\( g(x) = (2x)^2 = 4x^2 \)[/tex] results from a horizontal compression of [tex]\( f(x) = x^2 \)[/tex] by a factor of 2, the correct statement is:
C. The graph of [tex]\( g(x) \)[/tex] is horizontally compressed by a factor of 2.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
