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First graph f(x)= 1/4x^3 with points (0,0), (-3,-6 3/4), (-2,-2),(-1,-1/4), (1,1/4),(2,2) and (3, 6 3/4)

Second graph f(x)= 1/2 square root x
With points (0,0), (1, 1/2), (4,1), and (9, 1 1/2) describe the transformation
Both graphs represent vertical compressions with same factors of k
Represent vertical stretches, but different fatigue k
Vertical stretches with same k factors
Vertical compressions but different factors of k

1 answer

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:

  1. 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.

  2. 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.

  3. Vertical Stretches with Same \( k \) Factors:

    • Neither graph demonstrates vertical stretches, so there cannot be vertical stretches with the same factors.

  4. 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 \( k \).