find the formula for the graph of function f(x) given in the graph above

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24-10-2017
Mathematics

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find the formula for the graph of function f(x) given in the graph above

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mreijepatmat mreijepatmat · Answer 1 · 2017-10-24T22:03:57+02:00

This is a sin function since for x = 0, y = 0

The general formula of such functions is:

f(x) = A sin (bx + c) + d, where:
A= period
b = coefficient of x
c= phase shift (horizontal shift)
d= vertical shift

Since there is no vertical shift nor horizontal shift, our function becomes:

f(x) = A. sin(b.x)

In the graph we notice that A=2
Let's calculate b. Remember that the PERIOD = 2π/b, but the period given on the graph is 8π, then:
:
8π = 2π/b → and b= 2π/8π → b= 1/4

Finally the equation is : f(x) = 2.sin(x/4)

astha8579 astha8579 · Answer 2 · 2022-01-24T08:04:30+01:00

The given graph is scaled form of trigonometric function graph.

The formula for the graph function f(x) is given by:

[tex]f(x) = -2sin(\dfrac{x}{4})[/tex]

How to find the formula for f(x) for given graph?

Note down the properties one by one.

Firstly its a periodic function.

Secondly, the maximum value of the function is 2 and minimum value is -2

Thirdly the function assumes maximum or minimum value when it reaches to [tex]2 \pi n; n \in \mathbb Z[/tex]

Fourthly, the graph from the center to right side goes down then up then down then up and so on.

If you've seen graph of sine function, then you'd know that the graph is periodic(repeating same values after some interval), the maximum and minimum values being 1 and -1 and maximum or minimum values are achieved when input is [tex]\dfrac{n\pi}{2}; \: n \in \mathbb{Z}[/tex]

Plus, the graph from center to right goes up then down then up then down and so on.

Transforming sine graph to make the given graph:

First we need down then up instead of up and down. Since down is negative output in its graph and up is positive, thus we will negate the value of sine function.

[tex]-sin(x)[/tex] will have flipped mountains and valley's in comparison to that of sin(x).

Now since the maximum and minimum values are needed to be 2 and -2 instead of 1 and -1, thus we will scale the output with factor of 2.

[tex]-2sin(x)[/tex] will have maximum and minimum as 2 and -2 respectively.

We need the graph to be maximum or minimum at x = [tex]2 \pi n; n \in \mathbb Z[/tex] and not at x = [tex]\dfrac{n\pi}{2}; \: n \in \mathbb{Z}[/tex]. Thus, we will scale down the input of -2sin(x) so that it treats [tex]2 \pi n; n \in \mathbb Z[/tex] as if it was given [tex]\dfrac{n\pi}{2}; \: n \in \mathbb{Z}[/tex].

The scaling needs to be by 1/4 since [tex]\dfrac{1}{4} \times 2\pi n = \dfrac{n\pi}{2}[/tex]

Thus, the resultant function is [tex]-2sin(\dfrac{x}{4})[/tex]

The below plotted graph shows graph of sin(x) (in red) and that of [tex]-2sin(\dfrac{x}{4})[/tex] (in blue).

Learn more about graph of sine function here:

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