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The vertex of a parabola is the point where the curve changes direction. In the given problem, the function is written as \( f(x) = -2x^2 \). This is a simple form where the squared term is not shifted horizontally or vertically. Thus, both the horizontal (h) and vertical (k) shifts are zero. The general form of a parabola is \( f(x) = a(x-h)^2 + k \). Here, \( h \) and \( k \) set the vertex of the parabola. When \( h = 0 \) and \( k = 0 \), the vertex is at (0, 0). For our problem, the vertex is thus clearly at the origin. This point is crucial for graphing, as it helps anchor the parabola on the coordinate plane.

The axis of symmetry is a vertical line that runs through the vertex of the parabola. It divides the parabola into two mirror images. For the equation \( f(x) = -2x^2 \), the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \). In this case, the formula simplifies because \( b = 0 \). Therefore, the axis of symmetry is simply \( x = -\frac{0}{2(-2)} = 0 \). This means the vertical line passing through \( x = 0 \) (the y-axis) is our axis of symmetry. This line helps understand the symmetrical nature of the parabola, making graphing easier.

The domain of a parabola refers to all possible x-values (\( x \)) for which the function is defined. For any quadratic equation like \( f(x) = -2x^2 \), the domain is all real numbers. This is because you can input any real number, and the function will output a corresponding y-value.

The range of a parabola deals with the y-values (\( y \)) the function can output. For \( f(x) = -2x^2 \), the parabola opens downwards since the coefficient of \( x^2 \) is negative. Therefore, the y-values get larger as the x-values approach 0, peaking at the vertex before decreasing towards negative infinity. The highest y-value occurs at the vertex, which is at (0,0). Hence, the range is from negative infinity to zero, inclusive. Formally, we can write the range as \( (-\infty, 0] \). Understanding the domain and range ensures you know the extent of the parabola on the coordinate plane.