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.
