Question

Use the description to write the quadratic function in vertex form.

Answer 1

The parent function f(x) = x2 is vertically stretched by a factor of 2 and translated 3 units left to create g.

g(x) = ?

Answer 2

@mathmale

Answer 3

Remember that parabola I drew for you? First, the parabola y=x^2. Then, the same parabola shifted 1.5 units upward. Have you any idea how to represent the shifted parabola, that is, could you write the equation for it, given that

Answer 4

y=x^2 was the original equation of the original parabola?

Answer 5

I am helping someone solve a simple equation, I will be right back her in about 2 minuets!

Answer 6

I'll type more while you're thinking.
We're talking about horizontal and vertical translation of graphs / functions here.
Generally, if you have y=f(x) and you decide to translate the whole graph upward by c units, the equation becomes y=f(x)+c, where c can be either + or -.
If you have y=f(x) and decide to shift the whole graph left or right, the equation becomes y=f(x-c), where c can again be either + or -. Have you seen/heard this stuff b efore?

Answer 7

Ricardo, I'm willing to help you with several problems, one after the other, as I am doing now, but in return I do expect your full attention.

Answer 8

I have only poked the surface a couple of times,

Answer 9

So: Where were we? I'll need to go back and look at your latest question so as to get back on track.
Have you seen that stuff about translation before?

Answer 10

Yes, I have. Only a couple times, my lessons only show so much of one subject. Everything is really crammed together.

Answer 11

If you're interested, we could brainstorm on how to handle that workload.
But back to your question. You're given y=x^2. You want to translate the graph to the left by 3 units. Have you an idea, now, based on what I typed earlier, what the equation

Answer 12

of the shifted graph would look like?

Answer 13

Yes, let me see if I can make it real quick

Answer 14

Since this may be your first or near to first time, let me ask that same question by typing out the four possibilities:
y=x^2+3
y=x^2-3
y=(x+3)^2
y=(x-3)^2.
Please choose the one you think is correct, and explain why you chose it.

Answer 15

y = (x + 3)^2

Answer 16

These are superb graphs, and your equation is also correct. But why did you choose that particular equation over the other 3?

Answer 17

because if you just add or subtract 3 from x^2, it moves up or down the y-axis. As apposed to adding or subtracting 3 from x and then squaring that, your point would move left or right on the x-axis.

Answer 18

Horizontal translation:
If you're given y=x^2,
we can modify this equation to take into account horizontal translation.
If we translate the graph c units to the right, the equation becomes y=(x-c)^2.

You write very well.

Answer 19

Let me finish what I was typing:
If we translate the graph c units to the left, the equation becomes y=(x+c)^2.

Answer 20

So your answer is right on target. Congrats.

Answer 21

I have been an English Honors student for 3 years, I would hope my writing and grammar are presented as well. And thank you!

Answer 22

OK. Now please describe what you know about vertical streching/shrinking.

Answer 23

Either dividing or multiplying. So if I divided x^2/3, the line would widen, and if I multiplied x^2 * 3, the line would come closer together and compact.

Answer 24

Like so:

Answer 25

That's essentially correct. You discuss width; I'd discuss the height of the graph instead, but we're both still describing the graph correctly.
If you were to start with the graph of y=x^2, and then m ....

Wow, you're doing beautifully. Wonderful graphs.

Answer 26

Let me continue: ... start with the graph of y=x^2 and then multiply the right side by 2, what would the resulting graph look like? (Describe the result in terms of either height or width of the graph, as you wish.)

Answer 27

That graphing utility is a powerful tool. It enables you to experiment and learn quickly, visually, how any change to the equation of a function affects the graph of the function.

Answer 28

So: to recap:
Start with y=x^2.
Multiply the right side by 2: y=2x^2.
How will this change to the equation affect the appearance of the resulting graph?

Answer 29

The graph essentially would compact, the lines pull closer together.

Answer 30

Correct. I'd still urge you to use the wording: "The graph appears to have been stretched vertically by a factor of 2." We're discussing the topic of "vertical stretching/shrinking," which is why I propose the other wording.

Answer 31

Now let's put everything together to respond to the original qeustion you posed:

"The parent function f(x) = x2 is vertically stretched by a factor of 2 and translated 3 units left to create g."

Want to take a stab at answering that, based on our discussion so far?

Answer 32

Okay, i don't think that will be a problem. Thank you. So, because the graph stretched upward, it shrunk and came closer together, correct?

Answer 33

The new line would be placed at coordinates (-3, 0) and the new equation will be:

y = 2(x + 3)^2

Correct?

Answer 34

By the way, I'd suggest you type "x squared" as x^2, not as x2. " ^ " is universally understood to indicate exponentiation.
Just out of curiosity, have you experimented with the Equation Editor (b elow)?
\[y=x ^{2}, y=(x+3)\]

Answer 35

Yes I have\[y = x^2\]

Answer 36

Perfectly fine, except for terminology which could be better.
You're working with a curve, not a line, and it's the "vertex" that you're placing at (-3,0).
So:

This new function g(x) is y=2(x+3)^2, due to horizontal translation of 3 units to the left and vertical stretching that stems from multiplying x^2 by 2." The vertex of the resulting parabola is at (-3,0)."

Answer 37

Don't be put off by my pickiness. I'm a math prof, after all. Using the correct vocabulary will lead to your better understanding the problem involved and improved ability to explain your work to others.

Answer 38

Yes! Vertex! That is the word, for some reason I was thinking it was the Parabola, but I didn't want to type that down because I wasn't too sure of it.

Answer 39

Thank you! I don't see it as much as "pickiness" as I do "constructive criticism".

Answer 40

Because after all, you are helping me and I am learning from you.

Answer 41

So you're done with this problem. Any questions left to discuss? What a great attitude; thank you.

Answer 42

While you're thinking:
If you can afford it, you might want to consider buying reference books to keep on hand while you're studying online. A basic statistics book would have been helpful when we were discussing correlation and regression analysis. In my office I always had books immediatley at hand to check if I'd forgotten something, or to show students where they could learn more about the material.

Answer 43

Also: if you could afford it, consider buying a graphing calculator such as the TI-84 Plus. It's a very handy tool, and one that you might very well need and want in college.

Answer 44

Yes, this problem is finished. I have one more similar to this and another that is a word problem. And I definitely will look in to buying those materials. Unless I go to public school. That is why I am trying to finish the rest of my classes.

Answer 45

Think of yourself as Ricardo the Professor, neatly organized, with various math books, calculator immediately at hand, and knowing just where to look for needed info!

Answer 46

OK: Go ahead and post either of the two problems you still have left. I'm going out to get my daily newspaper, but will be right back.

Answer 47

Yes, this is the subject I do most poorly in. Right now I am pushing at a 76% C in the course and I am trying to finish with at least a B. But I only have 4 days until my deadline and I am about 80% done with the course.

Answer 48

I'll probably have more uninterrupted blocks of time in which to work with you. Please choose the problems you feel you have the best chance of understanding and warn me when you're about to post them.

Answer 49

Over the next several days, I mean.