Question

Analyze the graphs b(x)= 2/3√x - 9 +1
and k(x)=2/3√10/7(x - 9) + 1
, and determine the transformation that is occurring.(1 point)
Responses

horizontal compression by a factor of 0.7
horizontal compression by a factor of 0.7

horizontal compression by a factor of 10/7
horizontal compression by a factor of 10 sevenths

horizontal stretch by a factor of 7/10
horizontal stretch by a factor of 7 tenths

horizontal stretch by a factor of 0.8

Answer 1

To analyze the functions given, we will first rewrite both functions in a more standard form to identify the transformations applied.

1. The function \( b(x) \) is given by:
\[
b(x) = \frac{2}{3} \sqrt{x} - 9 + 1 = \frac{2}{3} \sqrt{x} - 8
\]

2. The function \( k(x) \) is given by:
\[
k(x) = \frac{2}{3} \sqrt{\frac{10}{7}(x - 9)} + 1
\]

To clarify the transformation from \( b(x) \) to \( k(x) \), we can examine how \( k(x) \) can be written in a way that reveals the horizontal transformation.

Focus on the argument of the square root in \( k(x) \):
\[
\frac{10}{7}(x - 9)
\]
This indicates that \( x \) has been subjected to a transformation. To analyze this transformation, we can compare it to the \( b(x) \) function.

### Step 1: Identify the factor

In the expression \( \frac{10}{7}(x - 9) \), we can write it as:
\[
\frac{10}{7}x - \frac{90}{7}
\]
This corresponds to a horizontal stretch or compression factor determined by the coefficient \( \frac{10}{7} \).

### Step 2: Find the transformation factor

In the case of :
- A transformation is of the form \( k(x) = f(mx + b) \) is performed, where \( m \) relates to horizontal scaling. If \( |m| > 1 \), there is a horizontal compression, while if \( |m| < 1 \), there is a horizontal stretch.

Here, compare this with \( b(x) \):
- It implies that the transformation applies a horizontal stretch by a factor of \( \frac{7}{10} \) because we see that the new \( x \) coordinates are produced from the original's stretched version.

### Conclusion

Thus, the transformation occurring between \( b(x) \) and \( k(x) \) is:
**horizontal stretch by a factor of 7/10.**