Determining Parabola Intervals of Increase and Decrease Using x² Coefficient and Vertex

To solve this problem, let's begin by analyzing the given pieces of information:

• The coefficient of $x^2$: For a parabola in the form $f(x) = ax^2 + bx + c$, this coefficient $a$ determines whether the parabola opens upwards ($a > 0$) or downwards ($a < 0$).
• The $x$-coordinate of the vertex: The vertex $x$-coordinate of the parabola is given by $x = -\frac{b}{2a}$. This point is crucial because it marks the transition point where the parabola shifts from increasing to decreasing or vice versa.

Based on the direction of the parabola determined by $a$ and the $x$-coordinate of the vertex, we can conclude:

• If $a > 0$ (parabola opens upwards), the function decreases on the interval $(-\infty, -\frac{b}{2a})$ and increases on the interval $(- \frac{b}{2a}, \infty)$.
• If $a < 0$ (parabola opens downwards), the function increases on the interval $(-\infty, -\frac{b}{2a})$ and decreases on the interval $(- \frac{b}{2a}, \infty)$.

Therefore, knowing both the coefficient of $x^2$ and the $x$-coordinate of the vertex allows us to determine the intervals of increase and decrease of the parabola.

Correct