Quadratic Functions
by Emily Kennedy

Consider the quadratic function

f(x) = ax² + bx + c
(Assume a ≠ 0)

The graph of this function is a parabola.

What effect does changing the values of a, b, or c (while holding the other two constant) have on the vertex of the parabola?

Let's find out!

First, we need to know the coordinates of our parabola's vertex.

We can see that the vertex lies on the axis of symmetry:

Let's find the equation for the axis of symmetry
for the parabola described by
y = ax² + bx + c

First, we complete the square:

This corresponds to taking the parabola described by y = x² and applying a horizontal shift units to the left, a vertical shift units up, and then a vertical stretch by a factor of .

The axis of symmetry of the parabola described by y = x² is the line x = 0 (since (-x)² = x²).

The stretch and vertical shift do not affect the axis of symmetry;
only the horizontal shift does.
So the axis of symmetry of our parabola is the line

Thus, the vertex of the parabola lies at

Now let's see what happens when we change a, b, or c.

Click here to continue.

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