# Parabola

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Hey guys, It's that annoying guy who asks too many questions again :-D I need some clarification on parabolas, I've done this in VCE already but I just can't remember how to determine the vertex... actually I do, its just I don't understand how to work out the signs (positive or negative). For example, the image above... clearly the k is -3 so shouldn't it move to the left? All the text explanations I've come across seems to be doing the opposite, seems if k is negative then shift right, and if k is positive then shift left. I surely am missing something here... what is it? Many thanks guys.

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What is x equal to if $y = (x-3)^2$?

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Had this in math class last month I hope my english in clear enough to explain it.

a graph is a visual representation of a function: if you put in a certain x in the function a y comes out*, the x and y value are the coordinates of a point on the graph.
with that in our mind we're going to look at the translated parabolas.
we put certain x in the first function say 5.
5^2 = 25 that means the point (5, 25) lays on the graph of that function.
now let's put 5 in the second function.
(5-3)^2 = 4 (2^2 = 4) that means the point (5, 4) lays on the graph.
now x = 8
(8-3)^2 = 25 (5^2 = 25) do you see that? 8-3 evaluates to 5 and gives 25.
That means that a point on the second graph with a x that is 3 higher than a point on the first graph gives the same y.
Translating a graph over the x-axis is indeed counter intuitive but when you get understand how it works it makes sense.

I hope this helps.

*not all function are valid for every x and don't give a y than ( y = 5/x for example is not valid for x = 0).

ps: keep in mind that translations over the y-axis are intuitive, so moving y = x^2 up 5 units becomes x^2 + 5.

[Edited by - flammable on March 2, 2010 10:22:02 AM]

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Quote:
 For example, the image above... clearly the k is -3 so shouldn't it move to the left?

k is not -3 here, it is +3. The equation is:

y = (x - k)^2 not y = (x + k)^2

here you can call k the axis of symmetry because when you draw graph, you draw it with respect to line x = k.

I hope this helped :)

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Y = (x - 3)^2

From the graph you see that the bottom of the parabola touches 0 when
x = 3. That is

Y(3) = (3 - 3)^2 = 0;

That means the parobala was shifted 3 units to the right, since the base
was shifted 3 units to the right.

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Quote:
 Original post by tenpoundbearFor example, the image above... clearly the k is -3 so shouldn't it move to the left?
No, k is +3 in that image. You're forgetting that subtracting a negative value is the same as adding a positive value.

The easiest way to think of it is to look at the roots (the places where y is zero).
With y = (x - k)^2, what value of x produces a y value of zero?:
0 = (x - k)^20 = x - k;k = x

Answer: the value of y is zero where x equals k.
Therefore if k is 3, it touches the x axis where x is 3.

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