Step by step
The question
Question: graph y = −3 cos x + 1.
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 −3, the number in front.
B is 1, the number in front of x.
Nothing is taken from x, so C is 0.
D is 1, the number added at the end.
2 Find the amplitude
The amplitude is how far the wave goes up and down from its middle line.
It is |A|: 3.
A is negative.
The minus flips the wave upside down: it starts by going down.
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
D moves the wave up 1.
Its middle line is y = 1, not the x-axis.
Careful!
The amplitude is from the middle line to the top, not from the bottom to the top.
Top to bottom is 6.
5 Make the key points
We use a table with five key points: one full wave.
x starts at 0.
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, plus D: times −3, then plus 1.
6 Draw the graph
Draw the grid.
The x-axis is in radians: a mark every π/2.
The middle line y = 1, dashed.
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 = 0.
Put it in the equation.
The inside is 0, and cos 0 is 1.
So y is −2, 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.