﻿ Translation and phase shifts of sine and cosine graphs. How equation relates to graph. Illustrated demonstrations and examples

# Translations of sine & Cosine Graphs

How the equation relates to the graphs

A translation is a type of transformation that is isometric (isometric means that the shape is not distorted in any way). A translation of a graph, whether its sine or cosine or anything, can be thought of a 'slide'. To translate a graph, all that you have to do is shift or slide the entire graph to a different place.

### Vertical Translations of Sine and Cosine graphs

This translation expresses a vertical shift downwards by 1.

Graph of f(x) = sin (-x) is the reflection of the graph of f(x) = sin (x) about x-axis.

Each pair of corresponding points on the graphs has the same distance form the x-axis.

For example, points A and B are two corresponding points on the graphs, and they are at the same distance from the x-axis. That is, AM = BM.

The difference between the variables of the two functions is

$$\left(x+\frac{5\pi }{4} \right)-\left(x+\frac{\pi }{4} \right)$$

= $$x+\frac{5\pi }{4} -x-\frac{\pi }{4}$$

= $$\frac{5\pi }{4} -\frac{\pi }{4}$$

= $$\frac{4\pi }{4}$$

= $$\pi$$

So, the graph of f(x) is transferred $$\pi$$ units to the left to create the graph of g(x).

The difference between the variables of the two functions is

$$\left(3x+\frac{\pi }{3} \right)-\left(3x-\frac{\pi }{3} \right)$$

= $$3x+\frac{\pi }{3} -3x+\frac{\pi }{3}$$

= $$\frac{\pi }{3} +\frac{\pi }{3}$$

= $$\frac{2\pi }{3}$$

Therefore, the graph of f(x) is transferred $$\frac{2\pi }{3}$$ units to the left to generate the graph of g(x)

Difference between the values of f(x) and g(x) is -3. Therefore, g(x) is obtained by moving f(x) 2 units downward.

First graph of $$f(x)=\sin (x)$$ is transferred to the graph of $$h(x)=\sin (x+2)$$ by moving 2 units to the left as shown in the graph below.

Then the graph of g(x) is obtained by moving the graph of h(x) upward 2 units

Graph of cos (-x) is exactly the same as the graph of cos (x).

So, g(x) is exactly the reflection of f(x) about x-axis.

$$g(x)-f(x)=\left[\cos \left(x-\frac{5\pi }{3} \right)-\frac{5\pi }{3} \right] - \cos \left(x-\frac{5\pi }{3} \right)$$

= $$\cos \left(x-\frac{5\pi }{3} \right)-\frac{5\pi }{3} -\cos \left(x-\frac{5\pi }{3} \right)$$

= $$-\frac{5\pi }{3}$$

So, the graph of f(x) is moved $$\frac{5\pi }{3}$$ units downward to generate the graph of g(x).

$$f(x)=\cos \left(\pi -x\right)$$ is the same as $$f(x)=\cos \left(x-\pi \right)$$. So $$g(x)=-\cos \left(x-\pi \right)$$ is the reflection of f(x) about x-axis.

$$g(x)=\cos \left(\frac{\pi }{3} \, \, -\, \, x\right)\, \, -\, \, \frac{\pi }{3}$$ is the same as $$g(x)=\cos \left(x-\frac{\pi }{3} \, \, \right)\, \, -\, \, \frac{\pi }{3}$$.
So, f(x) is moved downward $$\frac{\pi }{3}$$ units.

$$(3x+3\pi )-(3x-2\pi )$$

= $$3x+2\pi -3x+3\pi$$

= $$2\pi +3\pi$$

= $$5\pi$$

So, the graph of f(x) moved $$5\pi$$ units to the left in order to generate the graph of g(x).

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