Expert: Bobby Soltani - 1/14/2005

Question
What is the equation of y=x^2 + 4 after:
(a)a dilation of 2
(b)a rotation of 180 degrees
(c)a rotation of 270 degrees
(d)a reflection of y=x
(e)a reflection of y=-x
(f)a reflection of y=3
I'm taking a big test in a few weeks called the New York State regent and have to know this type of information.

Answer
Hi Jeff,

To answer this question, you should know the standard form of a parabola is

y=a(x-h)2+k

1.  a determines how quickly the parabola opens.  Larger a means that the parobola is more narrow.
2.  h,k is the x,y coordinates of the vertex.
3.  if a is negative, the parobola opens downward.
4.  If the parabola opens to the right instead of up, then the x and y variables are switched, and the h and k are switched.  Example:  a parabola that opens up is y=x^2.  The same parabola rotated 270 degrees clockwise opens to the left and has the equation x=-y^2.

Let's use these properties to solve the problem.
we have
y = 1(x - 0)^2 + 4
h = 0, k = 4

(a) change a to 1/2 to make it dilated by 2.  y =(1/2)x^2+4
(b) opens downward.  Change the x^2 to -x^2.
(c) opens to the left.  Change the x to -y and change the h and k.  x = -(y-4)^2 + 0
(d) y = x is a diagonal line that goes from the bottom left to the top right of a graph.  If we reflected the graph across this line, it would open up to the right and have its vertex at (4,0).  So we swap the h and k, and swap the x and y.  Sometimes it helps to check a couple of points to make sure you have the correct equation.  For instance, when y = 0, x should be 4 and it is.
x = (y - 0)^2 + 4
(e) same thing but it will open up to the left and vertex at -4,0
(f) opens downward and vertex at (0,2)

I hope you get the idea.  Let me know if you have questions.

Bobby