Let us consider the following quadratic function: f(x) = -x2 -6x -30

The solutions of the quadratic equation -x2 -6x -30 = 0 correspond to the zeros or the roots of the function f(x) = -x2 -6x -30.
These are the points where the graph of f(x) cuts the x-axis. There are no such points or roots in the graph of this function over here because the roots are complex, not real. The graph cuts the Y-axis at -30.

### The Discriminant and Roots of the Quadratic Equation -x2 -6x -30 = 0

The standard form of a quadratic equation is ax2 + bx + c = 0, where "a" does not equal 0. Note that if a = 0, the x2 term would disappear and the equation would be linear.
Looking at the given quadratic function  a = -1, b = -6, c = -30.
The discriminant D = b2 - 4ac = -62 - 4 * (-1) * (-30) = -84.0
The roots of the equation are complex conjugates and they are  (-b - √D)/2a and (-b + √D)/2a
= (-(-6) - √-84.0)/(2(-1)) and (-(-6) + √-84.0)/(2(-1))
-3.0+4i and -3.0-4i

The discriminant is negative. Hence, the roots are imaginary. The quadratic curve does not cut the X-axis. The curve lies below the X-axis.

Graph of y = f(x) = -x2 -6x -30

### Geometric and Graphical interpretation: Curve Sketching

The function f(x) = -x2 -6x -30 is the quadratic function. The graph of any quadratic function has the same general shape. This shape is called a parabola. The location and size of the parabola, and how it opens, depend on the values of coefficients in the function. This parabola has a maxima point and opens downwards. The graph of the parabola is symmetric with respect to the vertical line passing through the vertex.

The x-coordinate of the vertex will be located at x = (-b/2a) = (-(-6))/(2*-1) = -3.0, and the y-coordinate of the vertex is -21.0 which we obtain by substituting the value x = -3.0 in -x2 -6x -30. This is the maxima value attained by the quadratic function f(x). The derivative of the function is 0 at this point. This point is a turning point or a stationary point.

The vertex is the lowest point on the parabola if the parabola opens upward (coefficient a > 0) and is the highest point on the parabola if the parabola opens downward (coefficient a < 0)

Check the plot of another quadratic curve here.