Let's analyze the given transformation problem step by step.

We are comparing $f(x) = \log_3(x)$ and $g(x) = -\frac{1}{4} \log_3(5(x-7)) + 3$. From this, we can deduce the transformations applied:

1. Vertical compression by a factor of $\frac{1}{4}$:

• The coefficient $-\frac{1}{4}$ outside the logarithm indicates a vertical compression by $\frac{1}{4}$.

2. Reflection across the x-axis:

• The negative sign in $-\frac{1}{4}$ causes the graph to reflect across the x-axis.

3. Horizontal compression by a factor of $\frac{1}{5}$:

• Inside the logarithmic function, the $5(x-7)$ term indicates a horizontal compression by a factor of $\frac{1}{5}$, as the multiplier is reciprocal for horizontal transformations.

4. Translation 7 units to the right:

• The $x-7$ inside the logarithmic function translates the graph 7 units to the right.

5. Translation 3 units up:

• The $+3$ outside the logarithmic function translates the graph 3 units upward.

Correct transformations:

• Vertical compression of $\frac{1}{4}$ and horizontal compression of $\frac{1}{5}$.
• Reflection across the x-axis.
• Translation 7 units right and 3 units up.

Would you like me to explain any of these transformations further or provide visual examples?

Follow-up Questions:

1. How do reflections affect the orientation of the graph?
2. Why do coefficients outside and inside logarithmic functions affect vertical and horizontal transformations differently?
3. What happens if we translate the graph to the left instead of the right?
4. Can this transformation sequence be applied to a different base logarithmic function (e.g., $\log_2(x)$)?
5. What if $+3$ were replaced with $-3$? How would the transformation differ?

Tip:

Always apply transformations in the order of reflections, stretches/compressions, and then translations for consistency.