Short Answer 
The function y = -4x – 36 can be transformed into y = -4(x + 9), indicating a vertical stretch by a factor of 2, reflection over the y-axis, and a horizontal translation 9 units to the left. These transformations affect the graph’s steepness, direction, and overall position on the x-axis.
Step 1: Understand the Original Function
We start with the original function which is given as y = -4x – 36. To analyze the transformations, we first need to rewrite this function to identify its components. We can factor out the -4 to make it easier to compare with the parent function.
- Rewrite the function: y = -4(x + 9)
- This helps us identify the shifts and scaling effectively.
Step 2: Identify Transformations
By comparing with the parent function y = x, we can identify the specific transformations that have occurred. The characteristics of these transformations provide crucial insights into how the function has changed.
- Stretch by a factor of 2: The negative factor indicates a vertical stretch, meaning the y-values are being multiplied by 2.
- Reflection over the y-axis: The negative sign in front of the 4 reflects the function across the y-axis.
- Translation 9 units left: The term -9 indicates a horizontal shift to the left by 9 units.
Step 3: Confirm Transformations
To confirm that these transformations accurately describe the function, we can summarize the transformations identified. Each aspect of transformation tells us about the overall graph’s orientation and position.
- We find that the function is stretched by a factor of 2, which affects the steepness.
- The reflection over the y-axis alters the direction of the function.
- Finally, the translation means every point moves leftward, resulting in a complete repositioning along the x-axis.