Analyze the graphs b(x)=1.5(√4−x)−1 and c(x)=9(√4−x)−1. Determine the change that occurs from b(x)b(x) to c(x)c(x).(1 point)

Question

Responses
vertical compression by a factor of 9

vertical stretch by a factor of 7.5
vertical stretch by a factor of 6

vertical stretch by a factor of 9

Answer 1

To analyze the change from \( b(x) \) to \( c(x) \), we start with the given functions:

1. \( b(x) = 1.5(\sqrt{4 - x}) - 1 \)
2. \( c(x) = 9(\sqrt{4 - x}) - 1 \)

Identifying the Transformation

Both functions have the same basic structure: they are both vertical transformations of the function \( \sqrt{4 - x} \). The difference between \( b(x) \) and \( c(x) \) lies in the coefficients multiplied with \( \sqrt{4 - x} \).

1. Coefficient of \( b(x) \): 1.5
2. Coefficient of \( c(x) \): 9

Finding the Stretch Factor

To determine the vertical stretch factor from \( b(x) \) to \( c(x) \), we can calculate the ratio of the coefficients:

\[ \text{Stretch Factor} = \frac{\text{Coefficient of } c(x)}{\text{Coefficient of } b(x)} = \frac{9}{1.5} \]

Now, calculating the ratio:

\[ \frac{9}{1.5} = 6 \]

Conclusion

The transformation from \( b(x) \) to \( c(x) \) involves a vertical stretch by a factor of 6. Therefore, the correct response is:

vertical stretch by a factor of 6.