Find the Equation of a Trigonometri...
Find the Equation of a Trigonometric Function Given Its Graph - Grade 12

Graph of y equal cos bx
(y = a cos bx)

The graph of y equal cos bx which is y = a cos bx where 'a' is the amplitude and 'b' affects the period of the graph. The period is the distance that it takes for the cosine curve to begin repeating again.
As we know that the period for cosine curve is 2π. So the relation between period (2π) and 'b' is given by
Period =$ \frac{2π}{b}$
As the value of 'b' increases the period decreases.

Steps to draw the graph of y equal cos bx

Step 1 : Obtain the values of a and b.
Step 2: Draw the graph of y = cos x and mark the points where it crosses X-axis.
Step 3: Divide the x- coordinates of the points where y = cos x meets X-axis by b and also mark maximum and minimum values of y = a cos bx as 'a' and '-a' on the Y-axis.
For Example : 1) y = 3 sin 2x
Here the amplitude = a = 3
Period =$ \frac{2π}{b}$
b= 2
⇒ Period =$ \frac{2π}{2}$
So the cosine curve cut the x-axis at (π/4,0), (3π/4,0),(5π/4,0),...
The graph will look like
2) y = 2 cos 2x
Here the amplitude = a = 2
Period =$ \frac{2π}{b}$
b= 2
⇒ Period =$ \frac{2π}{2} = π $
So the cosine curve cut the x-axis at (π/4,0), (3π/4,0),(5π/4,0),...

3) From the given graph, find the equation.
From the above graph, we can see that the graph is decreasing curve so its cosine curve.
So the equation will be y = a cos bx
Now here the amplitude = a = 1
To find the period,
one cosine cycle is at 2π/3, so
Period =$ \frac{2π}{b}$ = $ \frac{2π}{3}$
So b = $ \frac{2π.3}{2π}$
b = 3
So the equation of the above graph is y = 1 cos(3x).

Practice on graph of y equal cos bx

1) State the amplitude and period for y = 4 cos 2x, and graph the equation for the interval $-π\leq x \leq 2π $
2) Compare the amplitudes of y = 1/2 cos x and y = 2 cos x with the amplitude of y = cos x and sketch a graph of each on the same coordinate system for $ 0 \leq x \leq 2π $

11th grade math

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