# Finding an equation algebraically

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Hello everyone,

I was just wondering if someone could provide some information as to how you would find an equation from a graph. Example, I have a graph with data points plotted that fits a curve such as a sine or cosine wave. What steps do I need to take algebraically to isolate an equation that would fit that data?

Many, many thanks!

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To find the trigonometric functions from the graph, find the:

amplitude a

period (2*PI)/b

y-intercept d

shift c

Amplitude is the vertical scaling as compared to 1sinx and 1cosx

Period is the horizontal scaling as compared to 1sinx and 1cosx

Y-intercept has the same function in linear functions as in trigonometric functions

Shift is what moves the function in the x direction. A negative shift moves it in the right direction, and vice versa.

Once you have those values, substitute them into the following equations:

y=a*cos(bx+c)+d

y=a*sin(bx+c)+d

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Thank you for that information. I am still a bit confused though.

How exactly algebraically do you perform those actions to find A, B, C, D?

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For Amplitude a, when finding the equation for blue, because red is 1cos(x), blue is 3cos(x).

For period b, when finding the equation for blue, because red is cos(1x), blue is cos(3x).

Also shows the period change. At x=PI, red, cos(1x) had completed half of its revolution, while blue, cos(3x), has completed 1/6 of it's revolution

For shift d, when finding the equation for blue, because red is cos(x+0), blue is cos(x+2).this is because cos(x+2) is shifted 2 to the negative x-direction.

y-intercept c is quite simple, it is just the distance from y=0 shifted up as compared to cos x or sin x

If by algebraically you mean only using relationships found in the graph, try this:

use the amplitude you found above, and plug it in to "amplitude" to solve for A, the value in cos(ax) or sin(ax)

distance over a revolution = (2*PI)/b

where the "distance of revolution" for sin(1x) is 2PI, sin(2x) is PI, ect.,

and where b is B in the equation

the shift is the x-axis shift, and the y-intercept is just like in linear equations.

Again, see this page for more assistance. It explains it quite well.

hope this helped

Edited by 2401 I Hate Tangents
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