How does the graph of the transformed function g(x)=log5(4x-16) compare to the graph of its parent function f(x)=log5x

Answer 1

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.

1. 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.

2. 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 $x=4$.

3. 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 $y=-\mathrm{\infty }$ as $x$ approaches 4 from the right, similar to how $f(x)$ approaches $y=-\mathrm{\infty }$ 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.