# How to Graph 1 cos 2x: Steps & Tutorial

Instructor: Sharon Linde
This lesson shows a step by step process for graphing the 1 cos 2x, which is a periodic function. We'll show you how to find the maximum and minimum values, and the values of x when y is equal to zero.

## Setting Up the Problem

Leo has always been a wiz with engines. When he was 9, he fixed the weed-eater engine his dad was too frustrated to keep working on. He hangs out at the local mechanic down the street from his house, and he's even started to make some money off of a side business fixing lawnmowers, hedge-trimmers, and other small engines.

Ordinarily, Leo wasn't the biggest fan of the math classes he had to take in high school, but when his teacher mentioned the cosine function was useful in engine design and repair, he definitely perked up. How could this math, that looked both boring and difficult, help him understand engines better? So, Leo did something he almost never did in math class - he raised his hand and asked a question.

'Mr. Grand, you said something about engines, which I love to work on, but then I look up and you have some funny looking squiggle drawn on the board. How does that squiggle represent anything about an engine?'

Mr. Grand explains that the sine and cosine functions can represent the repetitive movement of the cylinders in a combustion engine or the rotation of an electrically driven engine. The amplitude, or height of the graph, indicate how much the engine is moving each cycle, and the period, or distance between peaks on the graph, describe how quickly the parts are rotating. He draws a 'squiggly line' on the board to explain what he is talking about:

For the sine and cosine functions, you need to know a few things to be able to graph any particular function:

1. Maximum and minimum values of y
2. Value of y when x=0
3. Value of x when y=0
4. Distance between repetitions of the cycle

These concepts were stated differently than Leo normally thought of them, but he recognized the usefulness of knowing these things when he was working on engines. He watched closely as Mr. Grand filled in a chart related to graphing the function:

y = 1 cos (2x).

Leo thought of y as the distance a single piston in his engine was from a neutral position and x as the number of degrees the crankshaft was turning.

x 2x cos (2x) 1 cos (2x)
0 0 1 1
45 90 0 0
90 180 -1 -1
135 270 0 0
180 360 1 1

## Solution

Using the information that Mr. Grand had put on the board Leo is able to draw the graph of y = 1 cos (2x):

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