William W. answered • 10/22/21
Math and science made easy - learn from a retired engineer
I'm assuming your question is referring to:
f(x) = -2x2 + 2x + 1
This type of function is called a quadratic function and is identified by the variable "x" raised to the second power. The shape of the graph is a parabola and looks like this:
The number in front of the "x2" is called the leading coefficient, in this case "-2". If the leading coefficient is positive, the parabola opens upwards like the one shown above. The tip of the parabola (at the bottom) is called the vertex and, for a parabola opening up, this is the smallest y-value on the graph, also known as a minimum. If the leading coefficient is negative, the parabola will be upside-down from the one shown and the vertex will be the largest y-value on the graph, or a maximum.
In this case, the leading coefficient is negative, so its vertex is a maximum.
To find the vertex, if you write the function as f(x) = ax2 + bx + c, where a, b, and c are the coefficients then the x-value of the vertex is -b/(2a). In this case a = -2, b = 2, and c = 1 so the x-value of the vertex is -2/(2•(-2)) = -2/-4 = 1/2. To find the y-value of the vertex, plug in x = 1/2 (or 0.5):
f(1/2) = -2(1/2)2 + 2(1/2) + 1 = -2(1/4) + 1 + 1 = -1/2 + 1 + 1 = 3/2 (or 1.5) so the vertex is at the point (0.5, 1.5).
The axis of symmetry is x = 1/2 (this is the line that goes right down the middle of the parabola).
To graph it, start with the vertex (0.5, 1,5) then the parabola will go down on both sides. I sugest you look at desmos to see the graph but it will look something like this:
