Step by step
The question
Question: graph y = cos(x + π/4).
1 Read A, B, C, D
Every cosine graph can be written like this: y = A cos(B(x − C)) + D.
Each letter does one job.
Here A is 1, the number in front.
B is 1, the number in front of x.
x minus C: C is −π/4.
Nothing is added at the end, so D is 0.
2 Find the amplitude
The amplitude is how far the wave goes up and down from its middle line.
It is |A|: 1.
3 Find the period
One full wave of cos x is a full turn: 2π long.
B squeezes the wave: the period is 2π over B.
B is 1.
So the period is 2π: one full wave every 2π.
4 Find the shifts
C moves the wave sideways.
x − π/4 moves it left: the wave starts at x = −π/4.
Careful!
A plus inside the bracket moves the wave left, not right.
At x = −π/4, the inside is 0: that is where the wave starts.
5 Make the key points
We use a table with five key points: one full wave.
x starts at −π/4.
Each next x is a quarter of the period further: π/2.
A quarter of 2π is π/2.
The cos part follows the same pattern in every cosine wave: 1, 0, −1, 0, 1.
Each y is A times that: the same numbers.
6 Draw the graph
Draw the grid.
The x-axis is in radians: a mark every π/2.
Plot the key points from the table.
Join them with a smooth wave: round at the top and the bottom, never pointed.
The wave keeps going the same way, both ways.
7 Check
Let's check one point from the table: x = −π/4.
Put it in the equation.
The inside is 0, and cos 0 is 1.
So y is 1, like in the table.
So the graph is right!
Let's look back.
First: read A, B, C, D.
Then: find the amplitude.
Then: find the period.
Then: find the shifts.
Then: make the key points.
Then: draw the graph.
Last: check.