Let's break down the equation $y = -2(x + 4)^2 + 2$ in terms of transformations, domain, range, and key points.

1. Transformations:

• Parent function: The parent function is $y = x^2$, which is a standard parabola.
• Horizontal shift: The $(x + 4)$ indicates a shift 4 units to the left.
• Vertical stretch: The coefficient $-2$ means the graph is vertically stretched by a factor of 2 and reflected over the x-axis (opens downwards).
• Vertical shift: The $+2$ outside the squared term means the graph is shifted 2 units up.

2. Domain:

The domain of any quadratic function is all real numbers: $\text{Domain}: (-\infty, \infty)$

3. Range:

Since the parabola opens downward (due to the negative coefficient), the maximum value is at the vertex. The vertex is at $(-4, 2)$, so the range is all values less than or equal to 2: $\text{Range}: (-\infty, 2]$

4. Vertex:

The vertex is given by the transformation of the parent function: $\text{Vertex}: (-4, 2)$

5. Five key points:

• Vertex: $(-4, 2)$
• Point to the right of the vertex: $(-3, 0)$
• Point to the left of the vertex: $(-5, 0)$
• Another point to the right: $(-2, -6)$
• Another point to the left: $(-6, -6)$

These points are calculated by plugging in various values of $x$ into the equation.

Would you like more details or explanations on any part? Here are 5 related questions that might help expand your understanding:

1. How does a vertical stretch affect the steepness of the parabola?
2. What happens if the coefficient of the $x^2$ term is positive instead of negative?
3. How do you find the axis of symmetry for a quadratic function?
4. Can you explain how the horizontal and vertical shifts affect the graph?
5. How do you calculate the distance between the vertex and other points on the parabola?

Tip: When identifying transformations, always start with the basic parent function and then apply shifts, stretches, and reflections one by one.