Let’s plot the points on the graph that satisfy . That is, for each value of plot .
Every quadratic function of the form has this same shape. However, the coefficients and alter the position and steepness of the parabola. Try different values of and on this applet.
What’s a parabola good for?
1. A projectile. The height above the ground at time after launch (of a rocket, basket ball, slingball, or any other object thrown or dropped) can be modelled with a parabola.
2. Design (Engineering) and Animation (Films). Pixar partnered with Khan Academy to produce the ‘Pixar in a Box’ series of lessons and tutorials. Where math meets visual and performing arts.
3. This prezi showcases some parabola in action and offers a summary of the whole unit.
Three Forms of a Quadratic Expression
A quadratic expression has three common presentations.
- General form:
- Factored form
- Completed square (or vertex) form
Let’s see the expression in these three forms:
- General form
- Factored form
- Completed square form
These three forms are equivalent. No matter what value of you might choose, the three forms will compute to the same value. Let’s check with
Here is the graph of the function . Note the point lies on the parabola.
In this unit, we find out
- how to graph a parabola beginning with any of the three forms;
- how to algebraically convert from one form of quadratic expression to another;
- how to solve a quadratic equation.
Multiplying out Brackets
It is assumed that you know how to multiply out brackets such as . If not, you can review multiplying polynomials here.
By multiplying out the brackets we can already convert from factored form to general form:
We can also convert from completed square form to general form:
Check out these pages for the reverse operations: