The graph provided depicts a parabola that opens downward, and we need to match it with one of the given quadratic equations.

The standard form of a parabola is:

$y = a(x - h)^2 + k$

Where:

• $a$ determines the direction and width of the parabola.
  • If $a < 0$, the parabola opens downward.
  • If $a > 0$, the parabola opens upward.
  • The magnitude of $a$ controls how "wide" or "narrow" the parabola is.
• $(h, k)$ is the vertex of the parabola.

Analysis of the graph:

• The vertex appears to be at $(-3, 3)$.
• The parabola opens downward, so $a$ must be negative.
• The parabola is relatively wide, so the value of $a$ should be small in magnitude (close to 0).

Given equations:

1. $y = -0.2(x + 3)^2 + 3$
2. $y = 0.2(x - 3)^2 + 3$
3. $y = 0.2(x + 3)^2 + 3$
4. $y = -0.2(x - 3)^2 + 3$

Matching:

• The vertex $(-3, 3)$ is matched when $(x + 3)$ is squared (since $x + 3 = 0$ at $x = -3$).
• The parabola opens downward, so $a$ must be negative, meaning we need $a = -0.2$.

Thus, the correct equation is:

$y = -0.2(x + 3)^2 + 3$

Would you like more details or have any questions?

Related Questions:

1. What does the parameter $a$ in a quadratic equation represent geometrically?
2. How does the value of $h$ in the equation $y = a(x - h)^2 + k$ shift the parabola horizontally?
3. What is the effect of the $k$ value in a quadratic equation on the graph's vertical position?
4. How do you find the axis of symmetry of a parabola from its equation?
5. What does the vertex form of a quadratic equation reveal about the graph?

Tip:

The sign of the coefficient $a$ in the quadratic equation always tells you whether the parabola opens upwards (positive) or downwards (negative).