A quadratic function is a polynomial equation where the highest power of x is 2.
It might look like this: y = 2x2 - 6x + 1

Quadratic functions result in graphs that look like the one at the left.

The shape is called a parabola. It is a conic section, along with ellipses, circles and hyperbolas.

This example is the simplest possible parabola;
it has the equation y = x2 or y = 1x2

We're going to learn how to graph a quadratic function, and complete the square to find out important information about the graph.

We'll start with the end result. A quadratic function, after you've 'completed the square' (more on that later) will look something like this:

y = ±a(x ± c)2 ± d

Each colour-coded element of the function means something, which we'll explain.

First, the resulting graph will be a parabola, because it's quadratic (highest power is 2)

  • The sign out front determines the direction of opening: + opens up, - opens down
  • The first coefficient determines how steeply it opens; bigger = steeper
  • The sign and the constant c tell you how far left (+) or right (-) the graph is shifted
  • The sign and the constant d tell you how far up (+) or down (-) the graph is shifted
  • The vertex is the lowest (or highest) point on the parabola. It has coordinates (c, d),
          where the sign of c is the opposite of what's in the equation bracket
  • The axis of symmetry is a vertical line at c, with equation x = ±c (and opposite sign again)
  • The domain of all parabolas is x ε R
  • The range will be:
          y ≧ d if the parabola opens up
          y ≦ d if the parabola opens down

    Consider the information above a reference guide. We'll explain everything with examples.

    example 1: y = +2(x - 3)2 - 4

    The basic parabola is y = 1x2 with vertex (0, 0), at the top of the page

    The +2 tells you that it opens up (+) and it's steeper than y = 1x2

    The -3 with the x tells you it shifts the parabola right 3. This shift is opposite to the sign in the bracket

    The -4 at the end tells you it shifts the parabola down 4. This shift is NOT opposite to the sign

    With these two shifts, the vertex moves from (0, 0) to (3, -4)

    Here is more information about the graph:

    		 The graph at the left, with vertex (3, -4), shows the axis of symmetry, which is a vertical line through the x coordinate of the vertex, with equation x = 3

    At the right, the domain is all Real numbers, since the parabola travels left and right to infinity.

    The range y≧ -4


    Let's look at some more examples ...


    page 1 | page 2 | page 3 | page 4 | page 5



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    Content, HTML, graphics & design by Bill Willis 2023

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