Graphing transformations

I'm taking a correspondence math course which provides very little instruction on, well, anything.

And I'm way out of my league when it comes to graphing transformations. I understand how to get new coordinates from a mapping rule, but I don't understand how one arrives AT the mapping rule. For example:

1/2(y-1)=(x+3)^2 apparently turns into (x, y) -> (x-3, 2y+1)

I'm told it has a vertical stretch of 2, horizontal translation of -3, and vertical translation of 1. I pretty much get those terms--stretch means, well, you stretch it, and translation means where it moves to.

The best I can figure out is that for some reason, your stretch becomes whatever times your multiplyer of x or y makes one, and your translation is the opposite sign of whatever is added to your x or y.

But I'm not sure this is correct, and if it is, I don't understand WHY.

And maybe when I can understand that, I can better answer the next two questions (how can you tell from the equation that the vertex is at (-3, 1) and how can you tell from the equation that the sketching pattern is over 1, up 2 from the vertex, over 2, up 8, and so on?)

We were supposed to have learned this stuff in grade ten, but I missed a lot of time due to illness, and I'm not even positive that they taught much of it at all.

Any help would be very much appreciated. It's not so much the answers I want as an understanding of the concept.

Thanks!

-Shannon

Just look at it one piece at a time.

You have to start with something before you can modify it. Normally, something very simple would be a good place to start.

y = x^2

This is a parabola with vertex at (0,0) and passing through (1,1) and (-1,1). Line of symmetry is x = 0.

½y = x^2

This is a parabola with vertex at (0,0) and passing through (1,2) and (-1,2). Line of symmetry is x = 0. See how it is stretched horizontally?

½(y-1) = x^2

This is a parabola with vertex at (0,1) and passing through (1,3) and (-1,3). Line of symmetry is x = 0. See how it is moved up one?

½(y-1) = (x+3)^2

This is a parabola with vertex at (-3,1) and passing through (-4,3) and (-2,3). Line of symmetry is x = -3. y-intercept is (0,19). See how it is moved left three?

Something like that...

That was very helpful, thank you.

To reinforce my learning, to find the mapping rule, do you use the values that would reduce the equation back to its y=x^2 form?

For instance, if it's 2y+10=(3+x)^2, would it be a vertical stretch of 1/2, vertical translation of -10, and horizontal translation of -3?

Assuming that's correct, if your first parabola is, say, y=(3+x)^2, would you then reduce the second equation to that, or to the y=x^2 as in the first equation?

Maybe I'm not making sense. Tell me if I'm not making sense and I'll try somehow to explain better what I mean.

-Shannon

Go back to whatever fundamental form makes sense to you.

Getting you to wave your arm in just the right fashion should not be the goal of your teacher, textbook, or learning forum. Getting you to understand what you are doing should be the goal.