Need help understanding these two questions please. The first one is compressed horizontally by 2, but my guide only shows me hortizontal by 1/d. So what am I multiplying by? And the other question is expanded vertically by 2 which I believe is multiply by 1/2 but why does the answer look like this?

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To expand horizontally with scale factor 2, change "x" to "x/2" in equation.

To compress horizontally with scale factor 2, think of it as expanding with scale factor 1/2, so change "x" to "x/(1/2)" -- which is same as replacing "x" with "2x".

So in first example, you are compressing with scale factor 2, so replace "x" with "2x"

(2x)^2 / 4 - (y - 3)^2 / 36 = 1

This simplifies to 4x^2 / 4, or x^2 as you wrote.

x^2 - (y - 3)^2 / 36 = 1

On graph, every point is transformed horizontally so its horizontal distance from the y-axis is 1/2 what it was originally.

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To expand vertically with scale factor 2, change "y" to "y/2" in equation.

To compress vertically with scale factor 2, think of it as expanding with scale factor 1/2, so change "y" with "y/(1/2)" -- which is same as replacing "y" with "2y."

In second example, you are expanding vertically with scale factor 2, so replace y with y/2

(x + 2)^2  + ((y/2) - 3)^2  = 25

Now simplify.

(x + 2)^2 + ((y - 6)/2)^2 = 25

(x + 2)^2 + (y - 6)^2 / 4 = 25

Divide both sides by 25.

(x + 2)^2 / 25 + (y - 6)^2 / 100 = 1
Ok. Its pretty simple.

For b. Compressed…2 means you multiply the input by 2 and be aware of square. Input here is x value so (2x)^2 /4… = 4x^2 /4 = x^2 . Everything else is remained the same.

For c. Must divide both sides by 25 first. Same thing but multiply the output by 1/2 which means …(y-3)^2 /(25)4 =1 so …(y-3)^2 /100 = 1.
Let's take the graph x².

y = x² gives you a lovely parabola, when you plot the y values against the X values. We will call this specific shape that we just drew the "familiar parabola"

What about y= (2x)²? Well, that will give you a very similar graph. We can work out what graph it gives by noting the following trick. If you plot y=(2x)² on graph with a y aixs and a (2x) axis, then you get exactly the familiar parabola

Now, to turn that y-axis/(2x)-axis into a y-axis/x-axis graph, you'll need to pull horizontally outwards. Stretching the familiar parabola into a graph that is 2 times wider.

Similarly if you had y= (X+1)²  you can plot your familiar parabola on the y-axis/(X+1)-axis. To turn the X+1 axis into an X axis we need to shift all the numbers sideways (to the right).