Introduction to Quadratic Functions

Let’s plot the points on the graph that satisfy y=x^2. That is, for each value of x plot (x,x^2).

applet link

Every quadratic function of the form y=ax^2+bx+c has this same shape. However, the coefficients a,b and c alter the position and steepness of the parabola. Try different values of a,b and c 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.

https://dewwool.com/20-examples-of-projectile-motion/

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.

pixarinabox

3. This prezi showcases some parabola in action and offers a summary of the whole unit.

Prezi link

Three Forms of a Quadratic Expression

A quadratic expression has three common presentations.

  1. General form:                                  ax^2+bx+c
  2. Factored form                                   a(x-r_1)(x-r_2)
  3. Completed square (or vertex) form  a(x-h)^2+k

Let’s see the expression x^2-4x-5 in these three forms:

  1. General form                    x^2-4x-5
  2. Factored form                    (x-5)(x+1)
  3. Completed square form     (x-2)^2-9

These three forms are equivalent. No matter what value of x you might choose, the three forms will compute to the same value. Let’s check with x=6

  1. x^2-4x-5\quad =6^2-4(6)-5\quad=36-24-5\quad=7
  2. (x-5)(x+1)\quad=(6-5)(6+1)\quad=1\times 7 \quad =7
  3. (x-2)^2-9\quad =(6-2)^2-9\quad=4^2-9=16-9\quad =7

Here is the graph of the function y=x^2-4x-5. Note the point (6,7) 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 (x+1)(x-5). If not, you can review multiplying polynomials here.

By multiplying out the brackets we can already convert from factored form to general form:

    \begin{align*}&(x+1)(x-5)\hspace{1.5 cm} \text{multiply out brackets}\\[10 pt]&=x^2-5x+x-5 \hspace{1.3 cm} \text{gather like terms}\\[10 pt]&=x^2-4x-5\end{align}

We can also convert from completed square form to general form:

    \begin{align*}&(x-2)^2-9\\[10 pt]&=(x-2)(x-2)-9 \hspace{1.5 cm} \text{multiply out brackets}\\[10 pt]&=x^2-2x-2x+4-9\hspace{1.1 cm} \text{gather like terms}\\[10 pt]&=x^2-4x-5\end{align}

Check out these pages for the reverse operations:

From general form to factored form

From general form to completed square form

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