The shape of a roller coaster is modeled by a polynomial function, R(x). Describe how to find the x-intercepts of R(x) and how to construct a rough graph of R(x) so that the engineer can predict when there will be no change in the direction of the coaster. You may create a sample polynomial of degree 3 or higher to use in your explanations. Help?

- anonymous

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- anonymous

How would I do this problem if I decided to use x^3?

- anonymous

@ranga @tom982

- anonymous

Help me please?

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## More answers

- anonymous

@AlexandervonHumboldt2 @Compassionate

- retirEEd

Need a polynomial a little more complex to explain this properly.
Say for example: x^3+x^2-17x+15

- anonymous

Okay

- retirEEd

Can you explain what the x-intercepts means to you?

- anonymous

they will be the points at which the line crosses the x-axis?

- retirEEd

Very good!

- anonymous

Thanks. So, how do we get the x intercepts?

- retirEEd

It also means where Y equals zero, so these are the roots of the function.

- retirEEd

Do you know how to find the roots of a function? By saying factoring?

- anonymous

set it to zero

- anonymous

so, 0 = x^3 + x^2 - 17x + 15?

- retirEEd

That is the second step, first you need to simplify the polynomial.
something like (ax+b) (cx^2+dx+e)
This a difficult polynomial to determine the first root, so it you have a graphing calculator try to plot the equation.

- retirEEd

It will look something like this.
|dw:1449503213740:dw|

- anonymous

i agree. where do we go from here?

- anonymous

Part of my problem is that I don't know how to put together my answer.

- anonymous

I want to explain how to find the x intercepts, but I don't know how to do that and also I also want to explain how to construct a rough graph of R(x) so that the engineer can predict when there will be no change in the direction of the coaster.

- retirEEd

If your interest the roots to the polynomial are -5, 1, and 3.
0 = (x-1)(x-3)(x+5)

- retirEEd

I already told you how to find the x-intercept.
Can you predict when the train change direction, from up to down or down to up.

- retirEEd

CORRECTED ENTRY
The QUESTION does NOT ask up how to find the roots, just what how to find the x-intercept (ie find the roots) and
how does the engineer know when the coaster changes directions (form up to down or down to up).
So what do you think about the second part of the question?

- anonymous

Oh okay thanks. so i would say to make y = 0 and then solve? for the second part, he knows when to change direction because of the positive and negative values of x?

- retirEEd

No on both questions.

- anonymous

What do you propose?

- retirEEd

You said earlier when asked about the x-intercept
MathHelpPls
they will be the points at which the line crosses the x-axis?
What a these points of the polynomial called besides the x-intercepts?
They are the ________ of the equation. (Fill in the blank)

- anonymous

roots

- retirEEd

Yes!
So how would you answer the first question?

- anonymous

Upon graphing the equation, x^3 + x^2 - 17x + 15, you will see the points at which the line crosses the x-axis, making these points the x-intercepts of the graph. These points are the roots of the equation.

- anonymous

Is this right?

- retirEEd

Extremely excellent answer!

- anonymous

Whew! now i'm getting somewhere! haha Now, for part 2, I don't understand what causes the graph to change, like go up and down.

- anonymous

are you still there? @retirEEd

- retirEEd

Yes. I just had to really think about your question.
The function causes the response of the graph.

- retirEEd

How can the engineer predict when there will be NO change in the direction of the coaster?

- anonymous

I think i might just have to simply state exactly that. It doesn't say to explain why the graph changes direction or would not change. For my final answer, would saying this sound good?:
Upon graphing the equation, x^3 + x^2 - 17x + 15, you will see the points at which the line crosses the x-axis, making these points the x-intercepts of the graph. These points are the roots of the equation. The function, x^3 + x^2 - 17x + 15, causes the graph to move up and down in a specific way unique to this function.
But, it seems like i'm missing something.

- anonymous

i don't know how he will be able to.

- retirEEd

Look at the drawing of the graph I inserted. What is the primary difference bewteen the lines AB and CD?
|dw:1449506017315:dw|

- anonymous

they are going opposite directions. they are perpendicular to eachother

- retirEEd

The first part is correct. They weren't meant to be perpendicular, but take doesn't matter and besides, that's not a difference, but an observation.
Say they are perpendicular. What is unique about perpendicular lines?

- anonymous

perpendicular slopes have opposite signs. The other "opposite" thing with perpendicular slopes is that their values are reciprocals

- retirEEd

Correct.
So what if the lines weren't perpendicular, only the first part of answer would be true.
What does that tell you about the graph of the polynomial?

- anonymous

If the lines weren't perpendicular, then the lines would be straight and therefore have the same signs?

- anonymous

So, the graph of the polynomial would have areas where the line wouldn't change because the points have the same sign in those areas?

- retirEEd

You got it.
I realize you are very smart and appear to be training or testing my tutoring skills. I appreciate that, since I am very new to this endeavor.
But now it is time to take my dog for a walk. Have a good day!

- anonymous

Okay. Thank you for helping me. I appreciate that you spent this much time! You have helped a tremendous amount!

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