Could someone show how to find the axis of symmetry, the focus, and the directrix of these parabolas. Thank you.
1.y^2=12x
2. y=4(x-2)^2
3. (x+2)^2=y-3
4. y= x^2+4x+1
Hi there, the axis of symmetry can be found by transforming a parabola into turning point form. When you know the turning point i.e (a,b) then axis of symmetry is simply x=a.
Your first equation is not a parabola. Equations 2 & 3 are pretty much in turning point form, equation 4 can be put into turning point form by completing the square.
http://en.wikipedia.org/wiki/Parabola
1. You have to consider two different types of paraboae:
a) vertical axis of symmetry. The general equation of such a parabola is
$\displaystyle 4p(y-h) = (x-k)^2$
where V(k, h) is the vertex of the parabola and p is the distance between vertex V and focus F. The axis of symmetry has the equation x = k.
to #2.:
$\displaystyle y=4(x-2)^2~\implies~4\cdot \frac1{16} (y-0) = (x-2)^2$
Therefore: $\displaystyle V(2, 0)$; $\displaystyle F\left(2, \frac1{16} \right)$ ; axis of symmetry x = 2
b) horizontal axis of symmetry. The general equation of such a parabola is
$\displaystyle 4p(x-k) = (y-h)^2$
where V(k, h) is the vertex of the parabola and p is the distance between vertex V and focus F. The axis of symmetry has the equation y = h.
to #1.:
$\displaystyle y^2=12x~\implies~(y-0)^2=4\cdot 3 (x-0)$
Therefore: $\displaystyle V(0,0)$ ; $\displaystyle F(3, 0)$ ; axis of symmetry: y = 0
The remaining examples have to be done in just the same way.
They were supposed to have given you worked examples in your book and in the classroom lecture!
To make up for this deficiency, try some online lessons. Study at least two, and make sure you "follow" (understand each step) in the examples displayed.
Once you have learned the basic terms and techniques, please attempt at least one of the other exercises. If you get stuck, you will then be able to reply with a clear listing of your steps and reasoning so far, so we can "see" where you're having trouble, and then provide intelligent assistance.
Thank you!