Question

Shifting a parabola... horizontal reflection

Answer 1

Explain how the graph of the given function is a transformation of the graph y = x^2

Answer 2

y = (2x - 6)^2 + 5

Answer 3

I need someone to explain this to me, because I'm just not understanding it. The answer is: up 5, right 6, and stretched vertically by a factor of 2.

Answer 4

There is no reflection, just horizontal scaling, and translation.
In the general case for a parent function f(x),
\(y=f(bx-h)+k\)
compresses/stretches f(x) horizontally by a factor "b", and translates to the right by h, and up by k.
when b>1, it's a reduction, when b<1, it's a stretch.
Here
b=2, h=6, k=5

Answer 5

According to my lesson, y = c*f(x) stretches the graph of f(x) vertically by a factor of c, whereas y = f(cx) shrinks the graph of f(x) horizontally by a factor of 1/c times as wide. Now, why would this be a vertical stretch instead of a horizontal one when the c is on the inside of the parenthesis?

Answer 6

Right, my bad. I meant a stretch.

Answer 7

So it should be a horizontal stretch then?

Answer 8

Good, so the more general case is
\(y=af(bx-h)+k\)
where a is vertical stretching (a>1) or compression (a<1).
Your parent function is f(x)=x^2.

Answer 9

when b>1, it's a horizontal shrink/compression.

Answer 10

B is greater than 1 in the example right? B would equal 2

Answer 11

exactly, a=1, b=2, h=6, k=5.
It's a good formula to remember, solves the stretching/shrinking & translation cases.

Answer 12

My answer book says the answer to this problem is that it's stretched vertically by a factor of 2. Why would it be a vertical stretch and not a horizontal one?

Answer 13

It sounds like your answer book is answer a different problem. Double check for mix-up or typos. If none, mistake in the answer book! :(

Answer 14

Urg. This has been perplexing me. It definitely says vertical reflection. The full answer is: The graph is shifted up 5 units and right 3 units, then stretched vertically by a factor of 2.

Answer 15

This is part of the lesson also, if it helps..

Answer 16

If you are referring to this problem:
\(y = (2x - 6)^2 + 5\)
make sure it is not the following:
\(y = 2(x - 5)^2 + 3\)
because this seems to be what the answer seems to correspond to.

Answer 17

Also, beware that horizontal shrink can be made equivalent to a vertical stretch.
For example,
\(f(x)=(2x-5)^2+3\) .............(1)
\(=2^2(x-5/2)^2+3\)
\(=4(x-5/2)^2+3\) .................(2)
(1) and (2) are equivalent,
but the answer for (1) is
shrink horizontally by a factor of 2, translate 5 to the right, and up 3.
(2) is
Stretch vertically by 4, translate to the right 2.5, and up 3.

Answer 18

See below for illustration. The two (red & green) lines are fused together.
http://prntscr.com/bh9tfy

Answer 19

@mathmate The problem is in fact y = (2x - 6)^2 + 5 which is why I'm so confused.

Answer 20

Does anyone know the answer to this?

Answer 21

The answer might have been for the twin problem:
y=4(x-3)^2+5...............(2)
which is mathematically equivalent to
y= (2x - 6)^2 + 5 .............(1)
(see my explanations earlier).
For (1), it is
- horizontal shrinking by a factor of 2, then translate 6 to the right, and 5 up.
For (2), it is
- vertical stretch of 4, translate 3 to the right, and up 5.

Convince yourself, if possible, that (1) and (2) are mathematically equivalent.

Answer 22

The answer was "The graph is shifted up 5 units and right 3 units, then stretched vertically by a factor of 2. Neither of the examples you gave had all of those components...

Answer 23

Gahh. Perplexing.

Answer 24

I have another one that's been confusing me too (same lesson). Write a function for the graph described as a transformation of y = x^2. y=x^2 experiences a shift left by 2 units, then a horizontal shrink of factor 1/2,then a shift down of 5 units. The answer I got was y = (2x+2)^2 -5 ... it was wrong. Can you guess what the right answer was?

Answer 25

"The answer was "The graph is shifted up 5 units and right 3 units, then stretched vertically by a factor of \(\color{red}{2}\). Neither of the examples you gave had all of those components..."
I thought I explained the same transformations as No. (2) above?
"For (2), it is
- vertical stretch of 4, translate 3 to the right, and up 5.
"
For the vertical stretch, the book answer made a mistake. You can double check with your teacher.

Answer 26

Okay, I'll ask the teacher. I understand how to do all the transformations but that threw me off. If you say it's a mistake, I'll take your word for it since it doesn't really make sense.

Answer 27

Yep, please do that. Bring both versions. The other is more like what the answer is expected to be, so (repeating above)
The answer might have been for the twin problem:
y=4(x-3)^2+5...............(2)
which is mathematically equivalent to
y= (2x - 6)^2 + 5 .............(1)
(see my explanations earlier).
For (1), it is
- horizontal shrinking by a factor of 2, then translate 6 to the right, and 5 up.
For (2), it is
- vertical stretch of 4, translate 3 to the right, and up 5.