The linear parent function $p(x) = x$ represents the simplest form of a linear equation, where the graph is a straight line with a slope of 1 passing through the origin.

For the function $q(x) = -2p(x + 1)$, let's break it down:

1. Parent function: $p(x) = x$, so $p(x + 1) = x + 1$.
2. Transformation: $q(x) = -2(x + 1)$, which simplifies to: $q(x) = -2(x + 1) = -2x - 2$ This is a linear function with:
   • Slope: $-2$
   • y-intercept: $-2$

How this affects the graph:

• Reflection: The negative sign reflects the graph across the x-axis.
• Vertical stretch: The factor of $-2$ means the slope is steeper (compared to the parent function $p(x) = x$).
• Translation: The term $x + 1$ shifts the graph 1 unit to the left.

Now, let's consider the options:

• The graph should be a straight line with a negative slope ($-2$), steeper than the parent function.
• It should pass through the point $(-1, 0)$ due to the leftward shift by 1 unit.
• The y-intercept should be $-2$.

Without seeing the actual graphs (A, B, C, or D), look for a line with these characteristics.

Would you like further clarification or more details?

Questions for deeper understanding:

1. What does the slope of a linear function tell you about its graph?
2. How does reflecting a function across the x-axis affect its graph?
3. How do you determine the y-intercept of a transformed function?
4. Why does adding 1 inside the function shift the graph to the left?
5. How would the graph change if the coefficient were positive instead of negative?

Tip: In transformations, terms inside parentheses affect horizontal shifts, while outside terms affect vertical shifts or stretches.