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.
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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).
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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.
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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.
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