1. **Problem Statement:** Identify intervals of increase/decrease, symmetry, domain, and range for the function $f(x) = 3x^2$. 2. **Function Type:** This is a quadratic function of the form $f(x) = ax^2$ where $a=3 > 0$. 3. **Symmetry:** Quadratic functions with even powers are even functions. Check symmetry: $$f(-x) = 3(-x)^2 = 3x^2 = f(x)$$ Since $f(-x) = f(x)$, the function is even and symmetric about the $y$-axis. 4. **Intervals of Increase/Decrease:** - The vertex of $f(x) = 3x^2$ is at $(0,0)$. - For $x < 0$, the function is decreasing because the slope is negative. - For $x > 0$, the function is increasing because the slope is positive. 5. **Domain:** Quadratic functions are defined for all real numbers: $$\text{Domain} = \{x \in \mathbb{R}\}$$ 6. **Range:** Since $a=3 > 0$, the parabola opens upward with minimum value at vertex $f(0) = 0$: $$\text{Range} = \{y \in \mathbb{R} : y \geq 0\}$$ **Final summary:** - Symmetry: Even function, symmetric about the $y$-axis. - Increasing on $(0, \infty)$. - Decreasing on $(-\infty, 0)$. - Domain: $\mathbb{R}$. - Range: $[0, \infty)$.