Question

I need help! I am stuck and I do not know how to do this... can someone please help me work through this...This is an essay question.. Idk where to begin...

Given the function f(x) = 3(x − 3)2 + 2, indicate the shifts that will affect the location of the vertex and explain what effect they will have. Use complete sentences.

f(x+4)
f(x) + 4
f(4x)
4•f(x)

Answer 1

@whpalmer4

Answer 2

\[y = f(x)\]We can make that puppy sit up and do tricks :-)

What happens if we add or subtract something from the result of a function?

Answer 3

I am really not sure.

Answer 4

If we have \(f(x) = x\), simple case, a straight line going through the origin and up and to the right with a slope of 1.

Answer 5

so if you add or subtract something from the final result of a function it determines if it moves up or down?

Answer 6

Now let's add 1 to the function:
\[y = f(x) +1 = x+1\]What does our new graph look like? What is the value of \(y\) at \(x = 0\)? \(x=1\), \(x=-3\)?

Answer 7

Yes, adding a positive value to the result translates the graph in the direction of positive \(y\), and adding a negative value translates in the direction of negative \(y\). Very good!

Answer 8

Now here's a trickier case: how do we shift right or left?

Answer 9

Uhhh now that I am really not sure of.... I do not even have a guess.

Answer 10

and is that x=0?

Answer 11

Let's make a table of values:

x y=f(x)=x y = f(x+1) = x+1
-------------------------------
-3 -3 -2
-2 -2 -1
-1 -1 0
0 0 1
1 1 2
2 2 3
3 3 4

Answer 12

Imagine if you plotted both of the right hand columns? what would you get?

Answer 13

What do you mean what would I get? Would I have a parabola?

Answer 14

okay, here are the two lists of points:
(-3,-3), (-2,-2), (-1, -1), (0,0), (1,1), (2,2), (3,3)
that's the first line

(-3,-2), (-2, -1), (-1, 0), (0, 1), (1,2), (2,3), (3,4)
that's the second line

Answer 15

aren't the y values in the y = f(x+1) column identical to the values in the y = f(x) column, except shifted up or down by one row?

Answer 16

ohhh yess. I see that.

Answer 17

Right. so the effect of adding or subtracting from the ARGUMENT of the function is to translate the graph to the left or the right.

Answer 18

Okay that makes sens

Answer 19

if we subtract a positive number from the argument, instead of getting a graph feature that normally takes place at x = 0, we'll end up with the graph feature that takes place at x = <number we subtracted>.

Answer 20

and if we add a positive number, instead of getting a graph feature that normally takes place at x = 0, we'll end up with the graph feature that takes place at x = -<number we added>. refer back to my 3 column table and convince yourself that I speak truth :-)

Answer 21

it's easy and logical for the up/down translation — adding makes it go up, subtracting makes it go down. You would think that the natural order of things would have adding to the argument shifting the graph to the right, but in fact, it shifts to the left!

Answer 22

Okay (:

Answer 23

This makes much more sense now (:

Answer 24

So, now we can translate our graph right/left, and up/down. What about combination? What if we want to shift \(y = f(x)\) 2 units up, and 3 to the left? What would the new function be?

Answer 25

would I have f(x+2)-3 ?

Answer 26

well, the right pieces, but not quite the right arrangement. 2 units up, 3 to the left means we have to add 2 to the result of the function, and add 3 to the argument.

Here's a plot of 3 parabolas. In blue, we have \(y = f(x) = x^2\).
In purple, we have \( y = f(x+3)+2 = (x+3)^2+2\)
and in olive, we have \(y = f(x+2)-3 = (x+2)^2-3\)

Answer 27

oh okay (:

Answer 28

Hey, I am sorry to leave while you are in the middle of helping me... because it is rude. But my mom just called me to go do something, so I will be back.

Answer 29

Again I am sorry.

Answer 30

No problem! I may continue on without you :)

Answer 31

Okay I am back and ready to learn! lol

Answer 32

Oh, words to warm my heart :-)

Answer 33

hahaha xp

Answer 34

Hey, I'm not kidding, you know!

So looking back at your answer, do you see where you went wrong?

Answer 35

Not really.... I honestly am not understanding this... I am confused. I swear I am trying to understand... but my brain does not want to.

Answer 36

Okay. Let's recap:

adding to the result of the function translates up/down:

\(y = f(x)+a\) looks identical to \(y = f(x)\), except it appears \(a\) units higher on the y-axis. If \(a\) is negative, then it will be a negative amount higher (read: lower :-)

Answer 37

That's the easiest case to understand, so let's make sure it is crystal-clear.

Answer 38

Okay so if you add to a function it translates upward and if you subtract it translates down?

Answer 39

yes, assuming we are adding or subtracting positive quantities.

Answer 40

if we are adding or subtracting negative quantities, it moves in the opposite direction...

\[y = f(x)+(-1) = f(x) - 1\]right?

Answer 41

Right.... I think I understand lol

Answer 42

sometimes I'll ask questions which are really quite simple, like that one :-) Don't be fooled into thinking that you're missing something more complicated if it seems too simple to be true :-)

Answer 43

\[y = f(x)+a\]translates the graph along the \(y\) axis by \(a\), with \(a>0\) moving the graph in the direction of increasing \(y\) values, and \(a<0\) moving the graph in the direction of decreasing \(y\) values.

What happens if \(a=0\)?

Answer 44

Does it stay the same?

Answer 45

Very good. That might qualify as one of those questions I was talking about :-)

Answer 46

Okay haha (: I understand this a little better.

Answer 47

Now, let's wrestle with the right-left business again.

\[y=f(x+a)\]translates the graph along the \(x\) axis by \(a\), with \(a>0\) moving the graph to the LEFT and \(a<0\) moving the graph to the RIGHT

Answer 48

Okay (:

Answer 49

Suppose the graph has some interesting feature at \(x=3\). Maybe it touches the \(x\) axis there, or has a vertex, or whatever. Something readily recognizable.

If we add \(a\) to the argument before we put it in the \(f(x)\) machinery (in other words, we calculate \(f(x+a)\) instead of \(f(x)\), that thing we previously saw at \(x=3\) is now going to appear over at \(x = 3-a\), agreed?

Answer 50

if \(a=0\), the graph stays exactly where it started, of course. But if we added \(a\), to get the same result out of the machine, we need to put in \(x-a\), so that we have \(f((x-a)+a) = f(x - a + a) = f(x)\)

Answer 51

Yes. That makes sense.

Answer 52

I'm going to repost your question here so I don't have to scroll quite so far back:

Given the function\( f(x) = 3(x − 3)^2 + 2\), indicate the shifts that will affect the location of the vertex and explain what effect they will have. Use complete sentences.

\[f(x+4)\]\[
f(x) + 4\]\[
f(4x)\]\[
4•f(x)\]

Answer 53

Okay (:

Answer 54

Okay, we've covered the first two types of transformations. Now let's do the last two types mentioned here:

\[y = f(ax)\]\[y = af(x)\]

Answer 55

Alright (:

Answer 56

Now, before we do it, any guesses as to what multiplying either the argument or the result is going to do to the graph? We've already got shifting in each of the compass directions covered, so it's not that. What else might we do?

Answer 57

ummm do you like double shift or something lol? I have no idea lol xp

Answer 58

Do you stretch the graph???? possibly??

Answer 59

yes, you stretch or compress it, very good!!!

Answer 60

Haha... I feel smart xp

Answer 61

Let's do some experiments. I like experiments :-)

Let's say we have a function \(y = \sqrt{1-x^2}\) which looks a bit scary, but is just the top half of a circle with radius 1, centered at the origin \((0,0)\)

Answer 62

Okay (:

Answer 63

What do you think will happen if instead of plotting \(y = f(x) = \sqrt{1-x^2}\) I instead plot
\[y = 2*f(x) = 2*\sqrt{1-x^2}\]?

Answer 64

Will the graph be stretched by 2?

Answer 65

stretched up*

Answer 66

Oh cool so I was right! xp

Answer 67

Here's one where I multiplied by 1/2 instead of 2

Answer 68

let's see if I can trick you. What will \[y = -2f(x)\] look like?

Answer 69

wait so it got compressed when it was multiplied by 1/2

Answer 70

Yep!

Answer 71

Oh okay (: Why?

Answer 72

well, everything was half as tall, right?

Answer 73

it's a linear stretching or compressing along the y-axis

Answer 74

Yes. Okay that makes sense (: ..... anyway, if it is multiplied by -2 it would it be compressed by 2?

Answer 75

if |a|>1, it's a stretch, and |a|<1 a compression

Answer 76

ah, you fell into the trap :)

Answer 77

multiplying by -2 stretches it by a factor of 2, and inverts it!

Answer 78

Oh wow xp this is complicated lol

Answer 79

I probably wouldn't have fooled you if I had just shown you \[y = -1*f(x) = -f(x)\]first. But where's the fun for me in that? :-)

Answer 80

Changing the sign of the result just flips the graph over the x-axis. What was up, now is down :-)

Answer 81

hahaha xp At least one of us finds math fun.... just out of curiosity are you a teacher or something?

Answer 82

Nah, I'm an engineer.

Answer 83

Oh haha well that explains why you are good at math then

Answer 84

I'm on a quixotic quest to show people who think math is hard/they can't do it that it really doesn't have to be, at least for the stuff you learn in junior and senior high school. It's just a matter of getting the right explanation, practice, and careful work. It's very rewarding to get someone to go from "I hate math" to "OMG, I can do this, it's so easy!" :-)

Answer 85

I see, I see xp .... Well I used to like math until my freshman year of High school when I got a really terrible teacher... and ever since math and I have not been friends.

Answer 86

The problem for many is getting the explanation that works for them. The poor math teacher has to teach what, 20-40 kids at once, has a limited amount of time, and probably has a sucky work environment, too. I get to pick individual students who seem interested in learning, I can take multiple shots at explaining something until I find one that clicks, and I have a really comfy chair in front of a big computer, and the kitchen is just a few feet away :-)

Answer 87

So, do you see why changing the sign inverts the graph, on top of whatever else we might be doing to molest that poor, innocent function?

Answer 88

No I do not really get why it is inverted

Answer 89

Well, let's look at the table we didn't make:

x y -y
-3 0 0
-2 1 -1
-1 2 -2
0 3 -3
1 2 -2
2 1 -1
3 0 0

If you plotted the y column, you get something that looks like the upper half of a circle.

If you plot the -y column, you get the same thing, except where the top one went up 3, this one goes down 3, and so on. Isn't that how you would describe inversion?

Answer 90

Ohhh yes... Okay I understand (:

Answer 91

Right. So now we've handled stretching/compressing in the up/down direction, we still need to do that on the right-left direction. I bet you can tell me how...

Answer 92

Does that involve stretching and compressing to the right and left?? Possibly?

Answer 93

yes. but what might we do to the formula?

here's a hint:

translate up/down: add to the result
stretch/compress up/down: multiply with the result
translate left/right: add to the argument
stretch/compress right/left: ???