Question

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

Answer 1

y = (-3x)^2

Answer 2

I'm having a bit of trouble with this lesson. Horizontal and vertical stretches and whatnot.

Answer 3

Is it \[-3x ^{2} or (-3x)^{2}\]

Answer 4

According to my lesson, y = c*f(x) stretches the graph of f(x) vertically by a factor of c when c>1.
y = c*f(x) shrinks the graph of f(x) vertically to c times as tall when c<1.
y = f(cx) shrinks the graph of f(x) horizontally to 1/c times as wide when c>1.
y = f(x/c) stretches the graph of f(x) horizontally by a factor of c when c>1.

I don't see a formula for y = f(cx) when c<1, which this seems to be.

Answer 5

It is (-3x)^2, the second one.

Answer 6

Your question would then simply be \[y=9x ^{2}\]
Since \[(-3x)^{2} = 9x ^{2}\]

Answer 7

So in this case c would be 9

Answer 8

Ah. Interesting

Answer 9

You're sure?

Answer 10

yeah i am

Answer 11

So the answer would be "The graph y = x^2 would stretch vertically by a factor of 9" or something along those lines?

Answer 12

Mind clarifying a few more @Prathamesh_M ?

Answer 13

I need to do the same thing with y = x^ - 5 and y = (x - 5)^2

Answer 14

Second one is simple.
For \[y=x ^{2}\] The vertex is the origin.
For \[y=(x-5)^{2}\] the only change is that the vertex shifts to (5,0)

Answer 15

I'm not sure about the first one though

Answer 16

So it moves 5 to the right? for the first one?

Answer 17

What would the difference between y = x^2 - 5 and y = (x - 5)^2 be?

Answer 18

For \[y=(x-5)^{2}\]

Answer 19

\[y=x ^{2}-5\] is the same as \[y+5=x ^{2}\]

Answer 20

which means that the vertex would be (0,-5) instead of (5,0). The plots are identical in all other aspects

Answer 21

Ah okay... so the first equation would move down 5 units? and the second would move right 5?

Answer 22

yeah exactly

Answer 23

Basically you can express all such parabolas in the form of
\[(y-y _{1})=4a(x-x _{1})^{2}\]

Answer 24

Thank you so much! I just have one more left that I'm really confused about. y = -2x^2

Answer 25

For which the vertex would be (x1,y1)

Answer 26

Is that vertex form?

Answer 27

yes

Answer 28

That last one would be a downward open parabola

Answer 29

so you flip it upside down and stretch it by a factor of two

Answer 30

'it' here refers to \[y=x ^{2}\]

Answer 31

In the lesson I was given this: y = c*f(x) shrinks the graph of f(x) vertically to c times as tall when c<1.

Would it mean the same thing to say: "The graph y = x^2 would shrink vertically to 2 times as tall" ?

I don't really understand that.

Answer 32

Do u mean for that last one?

Answer 33

Yes

Answer 34

Because y = -2x^2 seems to fit that criteria.

Answer 35

The graph actually shrinks only when modulus of c is less than one

Answer 36

when u see a negative sign it means its downward open

Answer 37

Isn't -2 less than one?

Answer 38

yeah but i said modulus of -2

Answer 39

which would be 2

Answer 40

so actually what u typed before is wrong it should have modulus of 'c'

Answer 41

Are you saying this: "The graph y = x^2 would shrink vertically to 2 times as tall"

should be -2 instead of 2?

Answer 42

so graph shrinks only when c is something like 1/5

Answer 43

Oh I gotcha..

Answer 44

it would then shrink by a factor of 5

Answer 45

Got it. That makes sense. So for this problem, The graph would stretch vertically (or horizontally?) by a factor of 2 and also open downward?

Answer 46

Would that also make it a vertical reflection?

Answer 47

@Prathamesh_M

Answer 48

yup it would look like a water image of the graph stretched by a factor of 2.