## 4. Exploring the parameters of the quadratic function

In Section 2, you saw that the basic quadratic equation has three parameters:

y = ax^{2} + bx + c

The parameter a has to be non-zero, and if it is positive the parabola has a minimum value and is open at the top, like a cup. If a is negative, the parabola has a maximum value and is open at the bottom, like a dome.

Increasing the value of a makes the parabola steeper, whilst decreasing a makes the parabola shallower. If we plot y = x^{2} and y = 0.25x^{2}, y = 0.5x^{2}, and y = 2x^{2}, the results look like this:

The shallowest parabola is y = 0.25x^{2}, and the steepest is y = 2x^{2}.

What effect does changing the other parameters have? The simplest one to understand is the constant, c, so we will keep b = 0 initially. For a given value of a, this simply acts as a vertical offset. If we plot y = x^{2}, y = x^{2} + 5 and y = x^{2} -5, the results look like this:

The effect of changing the value of c is the same for an inverted parabola (when a is negative).

If we turn our attention to b, the coefficient for the x-term, we see that the curve changes location. If c = 0, a positive value of b moves the parabola down and to the right, so that the minimum occurs at a higher value of x but a lower value of y. A negative value of b shifts the parabola down and to the left, so that the minimum occurs at a lower value of x and a lower value of y.

The following plots are for y = x2 - 2x, y = x^{2} and y = x^{2} + 2x:

In the case of an inverted parabola (a is negative), a positive value of b shifts the parabola up and to the right, that is the maximum has higher x- and y- values. A negative value of b shifts the parabola up and to the left, so that the maximum has a lower x-value but a higher y-value.

The following plots are for y = -x^{2} - 2x, y = -x^{2} and y = -x^{2} + 2x: