Question

Describe the transformations of​ f(x) that produce​ g(x).
​f(x)equals6 x​; ​g(x)equalsStartFraction 6 x Over 7 EndFraction plus 5
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Part 1
Choose the correct answer below.
A.
First there is a vertical compression by a factor of one seventh
. Then there is a vertical translation up 5 units.
B.
First there is a vertical translation up 5 units. Then there is a vertical compression by a factor of one seventh
.
C.
First there is a vertical compression by a factor of 5. Then there is a vertical translation up one seventh
units.
D.
First there is a vertical translation up one seventh
units. Then there is a vertical compression by a factor of 5.

Answer 1

To analyze the transformation from \( f(x) = 6x \) to \( g(x) = \frac{6x}{7} + 5 \), let's break it down step by step.

1. **Starting Function**: \( f(x) = 6x \)

2. **Looking at the Function \( g(x) \)**:
\[
g(x) = \frac{6x}{7} + 5
\]

3. **Identifying the Transformations**:
- **Vertical Compression**: The term \( \frac{6x}{7} \) indicates a vertical compression by a factor of \( \frac{1}{7} \). This is because multiplying by \( \frac{1}{7} \) makes the output smaller by this factor.

- **Vertical Translation**: The \( +5 \) at the end indicates that the entire function is shifted vertically upwards by 5 units.

4. **Order of Transformations**:
- The vertical compression happens first, adjusting the steepness of the line, followed by the vertical translation where the entire output is raised by 5 units.

Given these transformations, the correct answer is:

**A. First there is a vertical compression by a factor of one seventh. Then there is a vertical translation up 5 units.**