Question

what is the equation of the image y=x² that has undergone the following transformations:
-horizontal translation of 1 unit to the left
-dilation by a factor of 2 in the y direction
-vertical translation of 8 units down
could you please explain your answer for me, because im soo bad at this transformation stuff. thanks.

Answer 1

There are some properties about parabolas that you need to know. All parabolas can be written in the following form:\[y=a(x+b)^2+c\]If you hold b and c fixed and move a, you'll find that the parabola expands/contracts as you vary the magnitude of a. You can see it since, if you start with y=x^2, then y=2x^2 will inflate every result by 2. You should make two plots to see this.

So to answer the second part of your question, dilation by a factor of 2 is done by multiplying x^2 by 2; \[y=2x^2\]

Answer 2

Now, when you vary b, holding a and c fixed, you change where each y-value occurs, given a x-value. For example, if you have

y=x^2
and
y=(x+1)^2

you should find that y is zero when the first one has x-value 0, and when the second has x-value, -1. The minimum has been shifted/translated horizontally to the left by 1 unit, which is what you need.

Combining what we have so far, you need,\[y=2(x+1)^2\]

Answer 3

The last part is this: hold a and b fixed and vary c. You'll find that this shifts the parabola up and down, depending on the value. When c=0, the parabola lies on the x-axis. If you increase c, you shift it up, and if you decrease it, you'll bring the parabola down.

You need to do the latter; you need to shift the parabola down 8 units. So c=-8.

Putting all three parts together gives you your result:\[y=2(x+1)^2-8\]

Answer 4

okay sweet, thankyou sooo much :)

Answer 5

You can thank me by becoming a fan - I want to get my points up ;)

Answer 6

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well. keep it up, you're preeed' smart (:

Answer 7

cheers