Question

What transformations can be applied to this function: f(x)=-8(6x-13) -16

Answer 1

To analyze the transformations applied to the function \( f(x) = -8(6x - 13) - 16 \), we can break down the components of the function and identify what transformations each part represents. The function can be rewritten into a more familiar form, highlighting its transformations:

1. **Starting with the linear expression inside the parentheses**:
The term \( 6x - 13 \) can be seen as a transformation of the basic linear function \( g(x) = x \).
- The coefficient \( 6 \) indicates a horizontal compression by a factor of \( \frac{1}{6} \).
- The \( -13 \) indicates a horizontal shift to the right by \( \frac{13}{6} \).

2. **Applying the negative coefficient**:
The term \( -8 \) outside the parentheses means we apply a vertical stretch by a factor of \( 8 \) and a reflection across the x-axis. So the vertical transformation is:
- Reflect across the x-axis.
- Stretch vertically by a factor of \( 8 \).

3. **Vertical translation**:
The constant \( -16 \) at the end indicates a downward shift of \( 16 \) units.

Putting it all together, the transformations applied to the basic function \( g(x) = x \) to obtain \( f(x) = -8(6x - 13) - 16 \) are as follows:

1. Horizontal compression by a factor of \( \frac{1}{6} \).
2. Horizontal shift right by \( \frac{13}{6} \).
3. Reflection across the x-axis.
4. Vertical stretch by a factor of \( 8 \).
5. Downward shift of \( 16 \) units.

In summary, the complete transformation is:
- Start with \( g(x) \),
- Compress horizontally by \( \frac{1}{6} \),
- Shift right by \( \frac{13}{6} \),
- Reflect across the x-axis,
- Stretch vertically by \( 8 \),
- Shift downward by \( 16 \).