3. For each \( g(x) \), describe the transformation(s) from the base function \( f(x)=x^{2} \). \( \begin{array}{l}\text { a) } g(x)=\left(\frac{x}{3}\right)^{2}-6 \\ \text { b) } g(x)=5(-2 x-6)^{2}+1\end{array} \)

To describe the transformations of the base function \( f(x) = x^2 \) for each given function \( g(x) \), we will analyze the changes in the function step by step. ### a) \( g(x) = \left(\frac{x}{3}\right)^{2} - 6 \) 1. **Horizontal Stretch**: The term \( \frac{x}{3} \) indicates a horizontal stretch by a factor of 3. This means that the graph of \( f(x) = x^2 \) will be stretched away from the y-axis. 2. **Vertical Shift Down**: The \( -6 \) at the end indicates a vertical shift downward by 6 units. This means that the entire graph will move down 6 units. **Summary of transformations for part a**: - Horizontal stretch by a factor of 3. - Vertical shift down by 6 units. ### b) \( g(x) = 5(-2x - 6)^{2} + 1 \) 1. **Horizontal Shift**: The expression \( -2x - 6 \) can be rewritten as \( -2(x + 3) \). This indicates a horizontal shift to the left by 3 units (since \( x + 3 = 0 \) when \( x = -3 \)). 2. **Horizontal Compression**: The factor of \( -2 \) indicates a horizontal compression by a factor of \( \frac{1}{2} \) (the negative sign also reflects the graph across the y-axis). 3. **Vertical Stretch**: The coefficient \( 5 \) outside the squared term indicates a vertical stretch by a factor of 5. This means that the graph will be stretched vertically, making it narrower. 4. **Vertical Shift Up**: The \( +1 \) at the end indicates a vertical shift upward by 1 unit. This means that the entire graph will move up 1 unit. **Summary of transformations for part b**: - Horizontal shift left by 3 units. - Horizontal compression by a factor of \( \frac{1}{2} \). - Vertical stretch by a factor of 5. - Vertical shift up by 1 unit. These transformations describe how the base function \( f(x) = x^2 \) is altered to produce the functions \( g(x) \) in each case.