To understand how the graph of the transformed function \( g(x) = \log_5(4x - 16) \) compares to the graph of its parent function \( f(x) = \log_5(x) \), we can analyze the transformations applied to the parent function.
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Horizontal Scaling: The term \( 4x \) indicates a horizontal scaling. Specifically, because \( x \) is multiplied by 4, the graph will be compressed horizontally by a factor of \( \frac{1}{4} \). This means points on the graph of \( f(x) \) will move closer to the y-axis.
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Horizontal Shift: The expression \( 4x - 16 \) can be rearranged to \( 4(x - 4) \), suggesting a horizontal shift. Setting \( 4x - 16 = 0 \) gives \( x = 4 \) as the value where \( g(x) = 0 \) (the x-intercept). Therefore, the graph of \( g(x) \) is shifted to the right by 4 units. This means that the vertical asymptote, initially at \( x = 0 \) for \( f(x) \), is now at \( x = 4 \).
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Reflection: There are no reflections or vertical shifts since we haven't added any negative signs or constants outside the logarithmic function.
Summary of the transformations:
- Horizontal Compression: Compression by a factor of \( \frac{1}{4} \).
- Horizontal Shift: Shifted right by 4 units.
Key features:
- The vertical asymptote of the graph \( g(x) \) is now at \( x = 4 \) instead of \( x = 0 \).
- The graph will still approach \( y = -\infty \) as \( x \) approaches 4 from the right, similar to how \( f(x) \) approaches \( y = -\infty \) as \( x \) approaches 0 from the right.
- The point \( (1, 0) \) for the parent function \( f(x) \) becomes \( (4.5, 0) \) in \( g(x) \) after the horizontal transformations.
In conclusion, the transformed function \( g(x) = \log_5(4x - 16) \) is derived from the parent function \( f(x) = \log_5(x) \) by horizontally compressing and shifting right, resulting in a similar logarithmic shape but with its key characteristics (asymptote and intercept) changed accordingly.
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