Question

Which of the following transformations are used when transforming the graph of the parent function f left parenthesis x right parenthesis equals log subscript 5 baseline x to the graph of g left parenthesis x right parenthesis equals negative 2 log subscript 5 baseline left parenthesis x minus 6 right parenthesis? Select all that apply.
(1 point)
Responses

Shift the graph of f left parenthesis x right parenthesis 6 units down.
Shift the graph of Image with alt text: f left parenthesis x right parenthesis 6 units down.

Shift the graph of f left parenthesis x right parenthesis 6 units to the left.
Shift the graph of Image with alt text: f left parenthesis x right parenthesis 6 units to the left.

Reflect the graph of f left parenthesis x right parenthesis over the x-axis.
Reflect the graph of Image with alt text: f left parenthesis x right parenthesis over the x -axis.

Stretch the graph of f left parenthesis x right parenthesis vertically by a factor of 2.
Stretch the graph of Image with alt text: f left parenthesis x right parenthesis vertically by a factor of 2.

Stretch the graph of f left parenthesis x right parenthesis vertically by a factor of negative one half.
Stretch the graph of Image with alt text: f left parenthesis x right parenthesis vertically by a factor of Image with alt text: negative one half .

Answer 1

To analyze the transformations from the parent function \( f(x) = \log_5(x) \) to the transformed function \( g(x) = -2 \log_5(x - 6) \), we can examine the changes applied step by step.

1. **Horizontal Shift**: The expression \( x - 6 \) indicates a horizontal shift. Since \( (x - 6) \) is inside the function, it shifts the graph to the right by 6 units (not down or to the left).

2. **Vertical Reflection**: The negative sign in front of \( 2 \) indicates that the graph is reflected over the x-axis. This means that all y-values of the original function \( f(x) \) are multiplied by -1.

3. **Vertical Stretch**: The coefficient \( -2 \) indicates a vertical stretch by a factor of 2 after the reflection. The function \( g(x) = -2 \log_5(x - 6) \) means that after the reflection, the graph is stretched vertically by a factor of 2.

Combining these observations:
- The graph shifts **6 units to the right** (not down or left).
- The graph is **reflected over the x-axis**.
- The graph is **stretched vertically by a factor of 2**.

Thus, the correct transformations are as follows:

- Shift the graph of \( f(x) \) 6 units to the right. (Correct)
- Reflect the graph of \( f(x) \) over the x-axis. (Correct)
- Stretch the graph of \( f(x) \) vertically by a factor of 2. (Correct)

The options to select would therefore be:
- **Reflect the graph of \( f(x) \) over the x-axis.**
- **Stretch the graph of \( f(x) \) vertically by a factor of 2.**
- **(Note: "Shift the graph of \( f(x) \) 6 units to the left" is incorrect; it should be a right shift as explained.)**

You would not select any options mentioning shifting down or to the left. Also, the option about stretching by negative one half would be incorrect as we are stretching by a positive factor after reflection.