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

horizontal stretch by a factor of 0.8
horizontal stretch by a factor of 0.8

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

horizontal compression by a factor of
10
7

Answer 1

To analyze the graphs of the two functions \( b(x) = \frac{2}{3} \sqrt{x - 9} + 1 \) and \( k(x) = \frac{2}{3} \sqrt{\frac{10}{7}(x - 9)} + 1 \), we need to look at the transformation from \( b(x) \) to \( k(x) \).

1. **Identify the base function:**
The base function is the square root function, \( f(x) = \sqrt{x} \).

2. **Transformation of \( b(x) \):**
- The function \( b(x) = \frac{2}{3} \sqrt{x - 9} + 1 \) is vertically compressed by a factor of \( \frac{2}{3} \), translated horizontally 9 units to the right, and translated vertically 1 unit up.

3. **Transformation of \( k(x) \):**
- The function \( k(x) = \frac{2}{3} \sqrt{\frac{10}{7}(x - 9)} + 1 \) has a similar vertical compression of \( \frac{2}{3} \) and a vertical translation of 1 unit up.
- However, inside the square root, we see \( \frac{10}{7}(x-9) \). This can be rewritten to identify the transformation. The factor \( \frac{10}{7} \) in front of \( (x - 9) \) affects the horizontal stretch or compression.

4. **Transformation from \( b(x) \) to \( k(x) \):**
- The transformation involves the term \( \frac{10}{7} \), which means we're effectively scaling the input horizontally. The reciprocal of this factor \( \frac{10}{7} \) gives us the necessary information on how the function is stretched or compressed.

**Horizontal Stretch/Compression Calculation:**
- To express this horizontal transformation, we can say that for the function \( g(x) = \sqrt{(m)(x - h)} \), where \( m \) is the coefficient, the horizontal stretch/compression is controlled by \( m \) as follows:
- If \( m > 1 \), there is a horizontal compression.
- If \( 0 < m < 1 \), there is a horizontal stretch.
- Here, \( m = \frac{10}{7} \), which is \( > 1\). Thus, the transformation is a horizontal compression.

The compression factor is given by the reciprocal of \( \frac{10}{7} \), which is \( \frac{7}{10} \).

**Final Answer:**
The transformation occurring is a horizontal compression by a factor of \( \frac{10}{7} \), or equivalently, a horizontal stretch by a factor of \( \frac{7}{10} \).

Thus, **the correct response based on your options is**:
- Horizontal compression by a factor of \( \frac{10}{7} \).