the first question is
f(x)= (x+1)^2 -1 Do the following in order to get g(x)
1. 3 units up
2. Horizontally spans triple the distance
3. reflect over y axis
From what I understand.
g(x)= (X+1)^2 +2
(1/3(x+1))^2 +2
and then (-1/3(x+1))^2 +2
what I don’t get is when I graph and plug in the top, it doesn’t add up. I think I’m doing my reflection over the y axis wrong. Is the negative supposed to be in front of the x, making it -x instead of 1/3.
can somebody please show me the step by step process to get to the final equation.
Combinations of horizontal transformations are tricky; but you have made only one mistake. It is easy to avoid if you focus on the x; but that is hard to do with a function like this one.
Rather than work with the function as written, first work with the function notation itself. Do the things you did on f, rather than on the expression:
g(x) = f(-(1/3)x) + 3
Here we have replaced x with -(1/3)x, which reflects over the y axis and stretches it horizontally by a factor of 3, and also translates the graph up 3. This is just what you thought you did.
Now apply the definition of f to this, and observe how it is different from what you wrote. You will see that the (x+1) got in the way and confused you. Your changes acted on (x+1) rather than on x itself as they have to do.
the second question is even more worse.
y=|X| is translated 8 units down, translated 3 units left, vertical spread to 1/2, and then reflected over the y and x axis. It has to be done in the order it is given.
what I got is y= -1/2 (|-x+3|) -8
This one is a different issue, because the function as given is more straightforward. Here, it's a matter of order. Different books express it differently, but my way of thinking of it is that for vertical transformation you work "from the inside out", following the order of operations, but for horizontal transformations, you have to work "from the outside in", reversing the order. A good way to see how it goes is to literally transform the function step by step:
f(x) = |x|
translate down 8 units: subtract 8
g(x) = |x| - 8
translate left 3 units: replace x with (x + 8)
h(x) = |x + 3| - 8
compress vertically by factor of 1/2 (which I think is what you mean -- the wording is hard here!): multiply result by 1/2
i(x) = (1/2)(|x + 3| - 8)
reflect over y axis: replace x with -x
j(x) = (1/2)(|-x + 3| - 8)
reflect over x axis: multiply result by -1
k(x) = -(1/2)(|-x + 3| - 8)
Your made only one mistake. Do you see it? It's almost invisible, so that for a moment I thought you had it right.
You did the horizontal transformations in the right order, translating first and then reflecting -- and that's where people usually make a mistake. But because you were told to first translate vertically, and then compress and reflect (which is not the usual order), you have to first subtract 8, then multiply the entire expression (in parentheses) by -1/2.
