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This example is to description how to draw the graph of the quadratic function y = -2x2 - 8x + 1. First, we need to find the vertex of the graph of the quadratic function. To find the vertex of this graph, we need to use the formula of the completing the square.

y = -2x2 - 8x + 1
= -2(x2 + 4x) + 1
= -2(x2 + 4x + 4 - 4) + 1
= -2(x2 + 4x + 4 + 8 + 1
= -2(x + 2)2 + 9

Therefore, the vertex of the quadratic function is (-2, 9), the axis of symmetry of the graph is x = -2.

Note: If the quadratic function is written in the form y = a(x - h)2 + k, then the vertex is (h, k), the axis of symmetry is x = h.

The graph of the quadratic function y = ax2 + bx + x, where the coefficient a can not be zero, is the set of points (x, y) that satisfy the quadratic function y = ax2 + bx + c. Because the coefficient a is less than zero, so the graph is open downward. So x = -2 is the maximum value of the quadratic function. We let x = -2 be the middle value of the set of x.

x ... -4 -3 -2 -1 0 ...
y ... 1 7 9 7 1 ...

You can choose as many values of x as you like and calculate the value of y to get the set of points (x, y), then connect those points to get the graph of the quadratic function y = -2x2 - 8x + 1.