Apply translations, reflections and stretches to parabolas

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Edit   Delete - Last Modified By: BJO at 21/11/2013 11:52:56 AM

In order to get the Turning-Point form of a quadratic, you need to be able to take a standard-form quadratic equation,

      y = ax2 + bx + c

and convert it by completing the square to give something that is the correct "shape",

       y = a(x - h)2 + k

Review how to complete the square using this lesson: 

Edit   Delete - Last Modified By: BJO at 21/11/2013 11:38:23 AM
You might also like to look at this video:
Edit   Delete - Last Modified By: BJO at 21/11/2013 11:52:24 AM
You can try the following GeoGebra presentation.  See if you can work out what is happening.
Edit   Delete - Last Modified By: BJO at 21/11/2013 11:55:57 AM
If you haven't already installed GeoGebra, you can download it from the following location
Edit   Delete - Last Modified By: BJO at 21/11/2013 11:56:53 AM
NOTE: the installer is in a zipfile, which you will need to unzip, and then run.
Edit   Delete - Last Modified By: dsi at 13/12/2013 9:56:20 AM
Edit   Delete - Last Modified By: BJO at 21/11/2013 2:40:55 PM

See if you can draw a circle in GeoGebra.  Start with the equation of a circle:

      y2 + x2 = r2

How might you make this "translatable", in the same way as the turning point form of a quadratic?  What parameters (sliders) would you use? Save your work to a GeoGebra file, and Email it to your teacher.

Edit   Delete - Last Modified By: dsi at 13/12/2013 9:59:35 AM
Edit   Delete - Last Modified By: BJO at 21/11/2013 3:03:14 PM

At the finish of this learning bite, you should have an understanding of how to "stretch" a quadratic, and what the effects of of the following transforms are:

f(x) --> f(-x)

f(x) --> -f(x)

f(x) --> -f(-x)

You should be able to describe in words what happens. 

 

Edit   Delete - Last Modified By: bjo at 22/11/2013 2:52:28 PM
You can practice using turning-point form to draw quadratic graphs, as well as read the turning point from a graph, by doing the questions in Exercise 5C, in your MathsQuest 10 text book.  For extra practice, you may wish to work through worked examples 9 and 10 in chapter 5.
Edit   Delete - Last Modified By: BJO at 21/11/2013 2:59:43 PM

Think about any quadratic equation in the form y = ax2 + bx + c.  We can also write this in functional notation:

f(x) = ax2 + bx + c ("f is a function of x")

If we chose some values for a, b, c, we could plot the quadratic curve for f(x) by choosing some x values (e.g. ...-2, -1, 0, 1, 2...) and then substitute these into the equation (...f(-2), f(-1), f(0), f(1), f(2)...).

What happens when we change the a coefficient? 

What would happen if we then plotted f(-x) ?  What does this mean? Remember, in substitution, everywhere we see x in the equation, we substitute it with what we're intrerested in: in this case -x, so we would have

      f(-x) = a(-x)2 +b(-x) + c

or  f(-x) = ax2 - bx + c

What about -f(x)?

What would -f(-x) look like? 

We could either plot these new quadratics by hand, or by using suitable technology (GeoGebra perhaps?).  What is happening?