Understanding how transformations affect the graph of a function is crucial for understanding the relationship between [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. Given:
[tex]\[ f(x) = x^3 + 2x^2 + x \][/tex]
[tex]\[ g(x) = \frac{1}{4} x^3 + \frac{1}{2} x^2 + \frac{1}{4} x \][/tex]
To identify the graph of [tex]\( g(x) \)[/tex] based on [tex]\( f(x) \)[/tex], we need to analyze how the coefficients in [tex]\( g(x) \)[/tex] are derived from those in [tex]\( f(x) \)[/tex].
1. Compare the Coefficients:
- The coefficient of [tex]\( x^3 \)[/tex] in [tex]\( f(x) \)[/tex] is [tex]\( 1 \)[/tex]. In [tex]\( g(x) \)[/tex], it is [tex]\( \frac{1}{4} \)[/tex]. This is [tex]\(\frac{1}{4}\)[/tex] of the original coefficient.
- The coefficient of [tex]\( x^2 \)[/tex] in [tex]\( f(x) \)[/tex] is [tex]\( 2 \)[/tex]. In [tex]\( g(x) \)[/tex], it is [tex]\( \frac{1}{2} \)[/tex]. This is [tex]\(\frac{1}{4}\)[/tex] of the original coefficient.
- The coefficient of [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] is [tex]\( 1 \)[/tex]. In [tex]\( g(x) \)[/tex], it is [tex]\( \frac{1}{4} \)[/tex]. This is [tex]\(\frac{1}{4}\)[/tex] of the original coefficient.
2. Determine the Transformation:
- Each term in [tex]\( g(x) \)[/tex] is scaled by a factor of [tex]\(\frac{1}{4}\)[/tex] compared to [tex]\( f(x) \)[/tex].
Thus, the function [tex]\( g(x) \)[/tex] can be seen as [tex]\( f(x) \)[/tex] scaled vertically by a factor of [tex]\(\frac{1}{4}\)[/tex].
3. Graph Appearance:
- When a function is scaled vertically by a factor of [tex]\(\frac{1}{4}\)[/tex], its graph becomes "compressed" towards the x-axis. Peaks and troughs of the graph of [tex]\( f(x) \)[/tex] will be closer to the x-axis in the graph of [tex]\( g(x) \)[/tex].
In conclusion, the graph of [tex]\( g(x) \)[/tex] will have the same shape as the graph of [tex]\( f(x) \)[/tex], but it will be vertically compressed by a factor of [tex]\(\frac{1}{4}\)[/tex]. Therefore, the correct answer is the graph where every point on the graph of [tex]\( f(x) \)[/tex] is one-fourth as far from the x-axis in the graph of [tex]\( g(x) \)[/tex].
