Let's analyze the function $f(x) = -3(-x - 2)^2 + 3$ in terms of translation and scaling.

Step-by-step breakdown:

1. Parent Function: The basic form of this function is derived from the standard quadratic function $y = x^2$, which is a parabola.

2. Translation: The function $f(x) = -3(-x - 2)^2 + 3$ can be interpreted by rewriting the terms inside the parentheses: $f(x) = -3(-(x + 2))^2 + 3 = -3(x + 2)^2 + 3$

   • The term $(x + 2)^2$ represents a horizontal shift to the left by 2 units (because of the $x + 2$).
   • The $+ 3$ outside the squared term represents a vertical shift upward by 3 units.

3. Scaling: The coefficient $-3$ in front of $(x + 2)^2$ does two things:

   • The negative sign indicates a reflection across the x-axis (i.e., the parabola opens downward instead of upward).
   • The factor 3 causes a vertical stretch by a factor of 3. This means the parabola becomes narrower because it grows faster vertically.

Summary of transformations:

• Horizontal shift: 2 units to the left.
• Vertical shift: 3 units upward.
• Vertical stretch: By a factor of 3.
• Reflection: Across the x-axis (the parabola opens downward).

Would you like further details or have any questions?

Related questions:

1. How would the graph of $f(x) = 2(x - 1)^2 - 4$ be transformed compared to $y = x^2$?
2. What happens if the coefficient of the squared term is less than 1 (e.g., $f(x) = \frac{1}{2}(x - 3)^2 + 5$)?
3. How do you determine the vertex of a function in vertex form?
4. What is the effect of changing the sign inside the parentheses (e.g., $(x - h)^2$ vs. $(x + h)^2$)?
5. What role does the horizontal shift play in the axis of symmetry of a parabola?

Tip:

For quadratic functions, the vertex form $f(x) = a(x - h)^2 + k$ is useful to quickly determine the vertex $(h, k)$, as well as the direction of the parabola and any stretching or shrinking.