To find the graph of $q(x) = -2p(x + 1)$, let's break down how this transformation affects the linear parent function $p(x) = x$:

1. Starting with the parent function:
   The parent function $p(x) = x$ is a simple line with a slope of 1 that passes through the origin.

2. Transformation applied to the parent function:

   • Horizontal Shift:
     The expression $p(x+1)$ inside the function indicates a leftward shift by 1 unit (because adding inside the parentheses moves the graph in the opposite direction).
   • Vertical Reflection and Stretch:
     The factor $-2$ outside the function reflects the graph across the x-axis (because of the negative sign) and vertically stretches the graph by a factor of 2 (because of the 2).

Step-by-Step Transformation:

• The graph is shifted 1 unit to the left.
• The slope of the line changes from $+1$ to $-2$, making the line steeper and pointing downwards.
• The line will now pass through $(-1, 0)$ instead of $(0, 0)$ due to the leftward shift.

To identify which of the graphs $A$, $B$, $C$, or $D$ matches this transformation, you should look for:

• A line with a negative slope (since the reflection occurs).
• A slope of -2 (indicating the steepness).
• The line should be shifted 1 unit to the left.

Conclusion:

Now, look at each graph and see which one reflects these transformations.

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Do you have the graphs or can you describe them? If you have them in an image format, feel free to upload the image!

Would you like more details on how the transformations work, or do you have any further questions?

Here are 5 questions to consider next:

1. What effect does adding a constant inside the function have on a graph?
2. How does multiplying a function by a negative constant affect its graph?
3. How can you determine the slope of a line from its equation?
4. What are the differences between horizontal and vertical shifts in function transformations?
5. How would the transformation look if the factor outside was positive?

Tip: When shifting a graph horizontally, remember that changes inside the parentheses (like $x+1$) move in the opposite direction to what you'd expect!