To compare the graphs of the two functions [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = 7 x^2 \)[/tex], let's analyze the transformations involved.
1. Original Function [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = x^2 \)[/tex] is a basic quadratic function. Its graph is a parabola that opens upwards, with its vertex at the origin (0,0).
2. Transformed Function [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) = 7 x^2 \)[/tex] is another quadratic function, but multiplied by a factor of 7. This means each y-value of [tex]\( f(x) \)[/tex] is multiplied by 7 to get [tex]\( g(x) \)[/tex].
To understand how this multiplication affects the graph, let's consider some points:
- For [tex]\( x = 1 \)[/tex],
- [tex]\( f(1) = 1^2 = 1 \)[/tex]
- [tex]\( g(1) = 7(1^2) = 7 \)[/tex]
- For [tex]\( x = 2 \)[/tex],
- [tex]\( f(2) = 2^2 = 4 \)[/tex]
- [tex]\( g(2) = 7(2^2) = 28 \)[/tex]
- For [tex]\( x = -1 \)[/tex],
- [tex]\( f(-1) = (-1)^2 = 1 \)[/tex]
- [tex]\( g(-1) = 7((-1)^2) = 7 \)[/tex]
From these points, we can see:
- When [tex]\( x = 1 \)[/tex], the y-value in [tex]\( g(x) \)[/tex] is 7 times that in [tex]\( f(x) \)[/tex].
- When [tex]\( x = 2 \)[/tex], the y-value in [tex]\( g(x) \)[/tex] is also 7 times that in [tex]\( f(x) \)[/tex].
- This pattern holds for any value of [tex]\( x \)[/tex]: [tex]\( g(x) = 7 \times f(x) \)[/tex].
Conclusion:
The factor of 7 affects only the vertical stretch of the graph. Specifically, it multiplies all the y-values by 7, stretching the graph vertically by a factor of 7.
Therefore, the correct statement is:
D. The graph of [tex]\( g(x) \)[/tex] is the graph of [tex]\( f(x) \)[/tex] vertically stretched by a factor of 7.
