Step by step
The question
Question: graph y = 3 sin(x + 90°) − 1.
The x values are in degrees.
1 Read A, B, C, D
Every sine graph can be written like this: y = A sin(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.
x minus C: C is −90°.
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.
3 Find the period
One full wave of sin x is a full turn: 360° long.
B squeezes the wave: the period is 360° over B.
B is 1.
So the period is 360°: one full wave every 360°.
4 Find the shifts
C moves the wave sideways.
x − 90° moves it left: the wave starts at x = −90°.
Careful!
A plus inside the bracket moves the wave left, not right.
At x = −90°, the inside is 0: that is where the wave starts.
D moves the wave down 1.
Its middle line is y = −1, not the x-axis.
5 Make the key points
We use a table with five key points: one full wave.
x starts at −90°.
Each next x is a quarter of the period further: 90°.
A quarter of 360° is 90°.
The sin part follows the same pattern in every sine wave: 0, 1, 0, −1, 0.
Each y is A times that, plus D: times 3, then minus 1.
6 Draw the graph
Draw the grid.
The x-axis is in degrees: a mark every 90°.
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 90°, and sin 90° 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.