To analyze the transformations of the two graphs you provided, we'll consider their shapes and the vertical changes they represent.
First Graph: \( f(x) = \frac{1}{4}x^3 \)
This function is a cubic function, and you provided specific points:
- (0, 0)
- (-3, -6.75)
- (-2, -2)
- (-1, -0.25)
- (1, 0.25)
- (2, 2)
- (3, 6.75)
The vertical compression factor here is \( k = \frac{1}{4} \). This means that for any input value \( x \), the output \( f(x) \) is scaled down to one-fourth of the value it would normally be (relative to the standard cubic function \( x^3 \)). This results in the graph being flatter compared to the original cubic function.
Second Graph: \( f(x) = \frac{1}{2} \sqrt{x} \)
This function is a square root function, and you provided the following points:
- (0, 0)
- (1, 0.5)
- (4, 1)
- (9, 1.5)
The vertical compression factor for this function is \( k = \frac{1}{2} \). Similar to the first graph, this means the output \( f(x) \) is scaled down to half of the output value of the standard square root function \( \sqrt{x} \). The graph appears flatter than the original square root graph as well.
Comparing the Transformations:
-
Vertical Compressions with Same Factors of \( k \):
- The first graph (\( \frac{1}{4}x^3 \)) has a compression factor of \( k = \frac{1}{4} \).
- The second graph (\( \frac{1}{2}\sqrt{x} \)) has a compression factor of \( k = \frac{1}{2} \).
- The factors of \( k \) used are different (\( \frac{1}{4} \) vs \( \frac{1}{2} \)), so this does not represent vertical compressions with the same factors.
-
Vertical Stretches but Different Factors of \( k \):
- Vertical stretches occur when a function is multiplied by a factor greater than 1. Since both functions involve compression factors less than 1, they do not represent vertical stretches.
-
Vertical Stretches with Same \( k \) Factors:
- Neither graph demonstrates vertical stretches, so there cannot be vertical stretches with the same factors.
-
Vertical Compressions but Different Factors of \( k \):
- As discussed, both graphs are vertically compressed but with different factors (the first graph compresses more drastically than the second). Therefore, this statement holds true.
Summary
- The first graph shows a vertical compression with factor \( k = \frac{1}{4} \).
- The second graph shows a vertical compression with factor \( k = \frac{1}{2} \).
- Both graphs represent vertical compressions but with different factors of \( k \).