To sketch $g(x) = -2f(x + 5) - 3$ based on the given function $f(x)$, we need to apply a series of transformations to $f(x)$:

1. Horizontal Shift: The $(x + 5)$ term indicates a horizontal shift to the left by 5 units.
2. Vertical Stretch and Reflection: The factor $-2$ means that we reflect the function over the x-axis (because of the negative sign) and then apply a vertical stretch by a factor of 2.
3. Vertical Shift: The $-3$ at the end indicates a downward shift of 3 units.

To create the graph of $g(x)$:

• Take each key point on $f(x)$, move it left by 5 units, reflect it across the x-axis, stretch it vertically by 2, and then move it down by 3 units.
• Plot these transformed points and connect them accordingly to form the shape of $g(x)$ on the same grid.

Would you like a detailed walkthrough on transforming specific points?

1. What is the effect of each transformation individually?
2. How would the graph change if the coefficient of $f(x)$ was different?
3. How does reflecting over the x-axis affect the orientation of the graph?
4. What would happen if the horizontal shift was to the right instead?
5. How would you verify if the transformation was done correctly?

Tip: Always apply transformations in the correct order: start with horizontal shifts, then apply reflections and stretches, and finally make any vertical shifts.