Question

Can someone please explain to me how to figure this problem out my teacher said I did not explain enough thank you
the question was asking how y=x^2 and y=2(x+3)2–4 3 ways the graph changes. How does the value of a change the graph? How does the value of h change the graph? How does the value of k change the graph? I told her y=x^2 graph has axis of symmetry x = 0 whereas y=2(x+3)2–4 graph has axis of symmetry x=-3
but it wasn't a good enough answer I would love to understand this though so I would appreciate it if someone took the time to help me

Answer 1

\[y=x^2\]
\[y=2(x+3)^2-4\]

let's say that \(y = f(x)\) to start the discussion. What exactly \(f(x)\) is doesn't matter too much for much of the discussion.

Think of \(f(x)\) as a little machine. You put in \(x\) and out comes a number.

Answer 2

She told me to recall that the format for of a quadratic function is y=a(x-h)^2+k and each of the variables a, h and k tell you something but this whole thing is confusing so i'm not sure how to start.

Answer 3

Now, what if we add something to the output of the machine?

If we graph \(y = f(x) + 1\), the graph will look just like the graph of \(f(x)\), except each point will be shifted in the direction of positive \(y\) by 1 unit, just like as if we plotted \(f(x)\) on a transparent sheet, laid it down over our graph paper, and then shifted it up 1 unit along the \(y\) axis.

Make sense so far?

Answer 4

Yes that makes sense it basically all depends on the numbers you're plugging in? so like if it's a negative number its going to be shifted in the direction of negative y?

Answer 5

that's right, adding a negative number to \(f(x)\) (or subtracting a positive number) would shift the graph downward.

Answer 6

translation is the fancy term for shifting it without changing its shape.

Answer 7

so, here's our first trick in our bag of tricks:

if we have a function \(f(x)\), we can translate it up or down by adding or subtracting from the result of \(f(x)\). For example, \(y = f(x) + a\) produces a copy of \(y = f(x)\) shifted along the \(y\) axis by \(a\) units.

Answer 8

Now let's consider a different trick. What if instead of adding or subtracting, we multiplied?

\[y = f(x)\]\[y = a*f(x)\]

What happens to the graph if \(a = 1\)?

Answer 9

it would be positive one unit

Answer 10

on the y axis?

Answer 11

Suppose \(y = f(x) = x^2\). Does the graph of that look any different than that of \(y = 1*f(x) = 1*x^2\)?

Answer 12

It's identical, I hope you'll agree :-)

Answer 13

yes(: because isn't it just like multiplying it by itself so it's the same equation pretty much

Answer 14

What about if we used \(-1\) instead of \(1\)?

What is the relationship of \(y = -1*f(x)\) to \(y = f(x)\)?

Answer 15

it wouldn't be identical anymore.. I think because it foes to being a negative on the graph instead of positive

Answer 16

goes*

Answer 17

Isn't it just going to be the same graph, upside down?

Answer 18

yes.. i'm sorry I shouldnt have said identical it's just different on the graph

Answer 19

it's when you plug in more numbers when it's not identical anymore right? or am I wrong i'm sorry I truly do not understand this

Answer 20

Let's take a simple example. Consider the line \(y=x\). Slope is \(1\), and it goes through the origin. (0,0), (1,1), (2,2), (3,3), etc.

Now look at \(y = -1*x = -x\). Slope is \(-1\), still goes through the origin. (0,0), (1,-1), (2,-2), (3,-3), etc.

Isn't that the same graph, just inverted?

Answer 21

Yes I think so

Answer 22

Okay, so multiplying by \(-1\) inverts a graph, or reflects it across the \(y\)-axis, if you prefer.

Answer 23

What if we multiply by say 3?

That's going to give us a graph which is 3 times as tall, right?

Answer 24

Yes(:

Answer 25

And what about if we multiply by \(-3\)?

Answer 26

it would be inverted so on the graph it would reflect across the y axis as -3?

Answer 27

same graph just upside down

Answer 28

it's going to be 3 times as tall as \(f(x)\), and inverted.

Answer 29

Okay I understand that(:

Answer 30

Okay, so multiplying by a positive number scales the graph. I didn't cover the case of a positive number less than 1, but hopefully you can see that if we used say \(\frac{1}{2}\) as our multiplier, we'd get a graph that was half as tall. Multiplying by a negative number scales the graph and inverts it.

Answer 31

Now, here's the slightly tricky one. How do we shift the graph right or left along the \(x\) axis? Any ideas?

Answer 32

If we changed the "H" variable it would affect the horizontal position of the graph which is the x axis right? so we would plug that in i'm guessing

Answer 33

if we add or subtract to/from the argument of \(f(x)\), we translate the graph to the left or the right. Any feature of the graph which normally appears at \(x=0\) will instead appear at \(x = a\) if we graph \(f(x+a)\), right?

Answer 34

sorry, at \(x = -a\)!

Answer 35

see, I told you it was tricky :-)

Answer 36

This is so confusing but yes I think I understand what you're saying

Answer 37

Here we have 3 parabolas. in blue, we have \(y = x^2\). in purple, \(y = (x+2)^2\). in olive, we have \(y = (x-3)^2\)

Answer 38

I almost always have to work it out again for myself each time I explain it :-)

Answer 39

now, do you know about vertex form for a parabola?

\[y=a(x-h)^2 + k\]gives you a parabola with vertex at \((h,k)\), opening upward (like a bowl on a table) if \(a>0\) and downward if \(a < 0\)

Answer 40

the simplest case has \(h = k = 0\), putting the vertex at the origin. Like our blue parabola in the graph.

Answer 41

\(a\) scales and inverts the parabola, as we determined earlier. \(k\) shifts the parabola up and down. \(h\) shifts it left and right.

Answer 42

Okay that makes sense but don't we make A,K, and H a actual number like they represent something and isn't that what I have to figure out for
y=x2

y=2(x+3)2−4

Answer 43

cause you just graphed it correct? so basically if we changed any of the numbers that represent A,K AND H it would totally change the parabola

Answer 44

Basically what I should say to my teacher is y=x^2 graph has axis of symmetry x = 0 whereas y=2(x+3)2–4 graph has axis of symmetry x=-3 the 3 ways the graph can change is because of a scales and inverts the parabola, k shifts the parabola up and down. and h shifts it left and right. is that what she's asking for or is there way more to this problem that I am not understanding

Answer 45

are you still there? @whpalmer4

Answer 46

I have to log off because my parents need to use my laptop but I really thank you for helping me if possible do you think you could message me if I did not understand this correctly I need to give my teacher a explanation and i'm scared to give her the wrong answer so if I just gave you the wrong answer could you message me please how to do this again thank you and ttyl @whpalmer4

Answer 47

sorry, got tied up in another problem!

yes. your equation is
\[y = 2(x+3)^2 -4\]compare with vertex form:
\[y=a(x-h)^2+k\]
to make everything match, we have \[a=2, h = -3, k=-4\]
so our vertex is at \((-3,-4)\) which implies that \(x = -3\) is a line of symmetry. And because the y-coordinate of the vertex is \(-4\), we've shifted the graph down by 4 units. And finally, the \(a = 2\) means we've scaled the graph to be 2 times as tall.