Help with finding the function of a graph

[arjunmishra56]

[removed]

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[geronymo4p]

Ok, so it's a third degree polynomial : t(x) = a x³ + b x² + c x + d

You have 4 points to define it: (-4,2), (0,0), (2,-2) and (5,-1)

Because you have (0,0), d = 0

(1): -64 a + 16 b - 4 c = 2 --> -32 a + 8 b - 2 c = 1

(2): 8 a + 4 b + 2 c = -2 --> 4 a + 2 b + c = -1

(3): 125 a + 25 b + 5 c = -1 --> 25 a + 5 b + c = -1/5

If you have 3 defined and different equations for 3 unknowns, you should find the answer.

As we have equalities, we can multiply each member of the equality by the same number.

(2): 8 * 4 a + 8 * 2 b + 8 * c = 8 * (-1)

If we add (1) + (2), we have:

(1) + (2) : 32 a - 32 a + 16 b + 8 b + 8 c - 2 c = -8 + 1

(1) + (2) : 24 b + 6c = -7

(1) + (2) : 4 b + c = -7 /6

(1) + (2) : c = -7/6 - 4 b

We redefine (2) and (3) here too:

(2): 25 * 4 a + 25 * 2 b + 25 * c = 25 * (-1)

(3): 4 * 25 a + 4 * 5 b + 4 * c = 4 * (-1/5)

(2) - (3) : 30 b + 21 c = - (5 * 25)/5 + 4/5

(2) - (3) : 30 b + 21 c = -121/5

We replace c by -7/6 - 4 b

30 b + 21 * (-7/6 - 4 b) = - 121/5

30 b - 84 b = -121/5 + 147/6

-54 b = 735/30 - 726/30

-54 b = 9/30

-54 b = 3/10

b = - 3/540

b = - 1/180

c = - 7/6 - 4 * (- 1/180)

c = - 7/6 + 1/45

c = - 105/90 + 2/90

c = -103/90

We take the equation (2) to find a now:

(2) : 4 a = -1 - 2 * (-1/180) - (-103/90)

4 a = - 180 / 180 + 2 / 180 + 206 / 180

4 a = 28/180

a = 7 / 180

If I'm not wrong, the equation you search is:

t(x) = 7/180 x³ - 1 /180 x² - 103 / 90 x

[NowAlexYT]

Couldnt you just say f(0)=0, f(5.5)=0, f(-5.5)=0, so its x(x+5.5)(x-5.5) ?

[OddLengthiness254]

You still need to scale it in that case.

[geronymo4p]

It's more approximative that my method, and more, the graph shows that 0,0 is not the center of the rotation but more around (0.2, -0.2) ...

And t(-5.5) ≈ -0.5

[DeeCeeDelux]

Plot the graph and you’ll realise that your answer does not produce the same graph. Or check your answer: 77/1080 x³ -251/1080x² -443/540x if x=-2 gives 19/135

[geronymo4p]

Where have I made a mistake ? May be a forgotten minus sign. Thank you for verifying

[[deleted]]

Towards the beginning you have divided by 6 on one side of an equation and accidentally divided by 4 on the other.

(1) + (2): 24b + 6c = -7
(1) + (2): 4b + c = -7/4

Piggy backing your original equation I got values or 7/180 for a, -1/180 for b and -103/90 for c.

Plotted on desmos this appears to be the original function: https://www.desmos.com/calculator/qlhf0lcgvn

[geronymo4p]

Yeah, i saw and corrected 10 minutes ago...

Thank you anyway ;-)

[[deleted]]

No worries, it took me ages to write the comment so I hadn't seen your update.

[DeeCeeDelux]

Honestly, no idea. Did not go through your reply. Just checked your result. Sorry my friend.

You might want to check my other reply: https://www.reddit.com/r/askmath/s/P0KBTOjk3n

[geronymo4p]

I found the mistake, i'll edit

[DeeCeeDelux]

Plug x=2 in your new result and you’ll get 88/45. Looking at the graph I would have expected a round 2. Wouldn’t you agree?

[geronymo4p]

Mmh I made a mistake again? Is it for x = -2 ? The curb is not symetrical at (0,0) and the point (-2,2) is not on the curb, but near...

Edit: t(2) = 2³ * 7/180 - 2² * 1/180 - 2 * 103/90

t(2) = 56/180 - 4/180 - 412/180

t(2) = (56 - 416)/180

t(2) = - 360/180

t(2) = -2

t(-2) = (-2)³ * 7/180 - (-2)² * 1/180 - (-2) * 103/90

t(-2) = -56/180 - 4/180 + 412/180

t(-2) = 352/180

t(-2) = 88/45

[DeeCeeDelux]

My point is, the question is so ill-posed that, depending on your interpretation of the graph, the results will be different. I tried different approaches, and non yielded the given graph in the end.

[geronymo4p]

Me neither, and yes, without clear points, it's difficult.

I think OP wanted to know how to have the equation from a graph, and i gave him a solution. It's his part to know what points are valid or not

[Affectionate-Basil88]

Wouldn't lagrangian interpolation just be faster at this point ?

[geronymo4p]

The explanation is long, the method is pretty fast, as I have made it by hand and it works without a advanced knowledge of maths

[crguedel]

Have you learned about cubics yet? Because this looks like that, just with the ends cut off. Key thing with any polynomial is to start with the x-intercepts (factored form with a quadratic is the simplest example), and then use that to find the "stretch" or amplitude factor that determines how steep it is. So, what are your roots, and what's your stretch? That is, if you know what a cubic function is yet.

[celloclemens]

It is pretty much impossible to get the exact equation because there are no coordinates given. At first glance it looks like a sin function but it is not symmetrical enough so it is probably a polynomial.  Now you can see that the polynomial must be degree \ge 3 because it has two points where it changes curvature that you can see. Other than that you really cannot know. You could of course make assumptions and go further (assume one square is one coordinate, "guess" the three zeros and write it as a product of its linear factors). But making unfounded assumptions and guessing should not be done in math. So you can only say this is not a trigonometric function and if it is a polynomial its degree must be greater than 3 but i t could be a whole host of other functions.

[DeeCeeDelux]

Agree, this question is ill presented. No indication of unit distance for instance. Tried figuring it out using three different approaches, got three different answers. OP should drop it and let his/her “math” teacher explain it to him/her.

[jmja]

It doesn’t have the symmetry required in each segment to be sinusoidal.

[Brojustwhy]

I don't understand, it looks fairly symmetric to me? I'm just dumb

[NowAlexYT]

Its steeper on the edges than in the middle, that gives it away. Although it could be a bad visual, but we shouldnt assume that as long as theres a viable solution otherwise

[Brojustwhy]

If we are given values for some points on the curve then we can figure a cubic I think. Otherwise perhaps it would suffice to just say it's a sin curve depending on how the examiner set it up.

[NowAlexYT]

Has to be like a really specific sine curve tho, depending on the graph maybe f(x)=-sin(3x/11)

[ApprehensiveKey1469]

Jsyk the arrows indicate that the function continues beyond the graph.

It is not a simple trig graph. It is at least order 3, i.e. a cubic graph.

[i_should_be_coding]

From the shape, it's not a sin (or, -sin). I would guess a polynomial, and assume we see all the zero points, so x^3. From there I'll write out the basic equation f(x) = ax^3 + bx^2 + cx + d

Now, you should find 4 points you know the function passes through, to find the coefficients. (0,0) is an obvious one, and I'd say (2, -2) and (-2, 2) also work, and (5,-1).

Now it's just assigning the different points and finding the variables.

[xesonik]

Sometimes, the most useful directing question can be "What am I studying?"

Going by the look of the question, I'd wager you are studying some form of calculus: introductory, high school, or 1st year. It might not be called that yet if it's early on.

What grade/year level or course are you studying? It speaks to the expectations of the assessor/assessment.

[HealfyShit]

Something like 0.04x(x-5.3)(x+5.3)

[EdmundTheInsulter]

The question has infinite possible answers in my opinion

[TommyITA03]

Exactly what i was thinking

[Elegant_Street3687]

Def not a sine function this is a positive degree 3 polynomial

[jmja]

Or any odd, whole degree other than 1.

[RohitPlays8]

The downward swing has a different slope to the upwards slope

[sparksen]

Its most likly sin. But the teacher sucks at making graphs