Question

Determine the transformations that produce the graph of the functions g(x) = -3log(x - 8) - 9 and h(x) = -1/3 log(x - 8) + 9 from the parent function f(x) = log(x). Then compare the similarities and differences between the two functions, including the domain and range.

Answer 1

The transformations applied to the parent function
`\(f(x) = \log(x)\)` to obtain
`\(g(x) = -3\log(x - 8) - 9\)` involve a vertical compression by a factor of \(1/3\), a horizontal shift to the right by 8 units, a vertical reflection, and a vertical shift downward by 9 units. Similarly, for
`\(h(x) = -1/3 \log(x - 8) + 9\),` the transformations include a vertical stretch by a factor of 3, a horizontal shift to the right by 8 units, and a vertical shift upward by 9 units. The differences lie in the vertical stretches or compressions and the direction of the shifts.

Step-by-step explanation:

To obtain
`\(g(x)\),` the coefficient
`\(-3\)` indicates a vertical compression by a factor of \(1/3\) compared to the parent function
`\(f(x)\)`. The horizontal shift of 8 units to the right is denoted by
`\((x - 8)\)`, and the negative sign before the logarithmic term reflects the graph vertically. The final vertical shift downward by 9 units is represented by
`\(-9\)` outside the logarithmic term.

For
`\(h(x)\)`, the coefficient
`\(-1/3\)` implies a vertical stretch by a factor of 3. The horizontal shift to the right by 8 units is indicated by
`\((x - 8)\)`, and the positive sign before the logarithmic term reflects the graph vertically. The final vertical shift upward by 9 units is represented by
`\(+9\)` outside the logarithmic term.

The comparison between the two functions reveals that
`\(g(x)\)` has a vertical compression and downward shift, while
`\(h(x)\)` has a vertical stretch and an upward shift. These transformations affect the domain and range differently, altering the scale and position of the logarithmic graph.