Worked examples
The question
Question: graph y = 3 sin x.
y = 3 sin x
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.
y = A sin(B(x − C)) + D
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.
Nothing is added at the end, so D is 0.
A = 3
B = 1
C = 0
D = 0
2 Find the amplitude
The amplitude is how far the wave goes up and down from its middle line.
It is |A|: 3.
amplitude = |3| = 3
3 Find the period
One full wave of sin x is a full turn: 2π long.
B squeezes the wave: the period is 2π over B.
period = 2πB
B is 1.
So the period is 2π: one full wave every 2π.
period = 2π1 = 2π
Careful!
The amplitude is from the middle line to the top, not from the bottom to the top.
Top to bottom is 6.
Careful: amplitude = 3, not 6
4 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.
0
π2
π
3π2
2π
A quarter of 2π is π/2.
a quarter of 2π = π2
The sin part follows the same pattern in every sine wave: 0, 1, 0, −1, 0.
0
1
0
−1
0
Each y is A times that: times 3.
0
3
0
−3
0
5 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.
6 Check
Let's check one point from the table: x = π/2.
Put it in the equation.
x = π2: y = 3 × sin(π2)
The inside is π/2, and sin π/2 is 1.
So y is 3, like in the table.
So the graph is right!
= 3 × sin π2 = 3 × 1 = 3
Let's look back.
First: read A, B, C, D.
Then: find the amplitude.
Then: find the period.
Then: make the key points.
Then: draw the graph.
Last: check.
▶ Watch it solved
The question
Question: graph y = 2 sin x + 1.
y = 2 sin x + 1
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.
y = A sin(B(x − C)) + D
Here A is 2, 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.
A = 2
B = 1
C = 0
D = 1
2 Find the amplitude
The amplitude is how far the wave goes up and down from its middle line.
It is |A|: 2.
amplitude = |2| = 2
3 Find the period
One full wave of sin x is a full turn: 2π long.
B squeezes the wave: the period is 2π over B.
period = 2πB
B is 1.
So the period is 2π: one full wave every 2π.
period = 2π1 = 2π
4 Find the shifts
D moves the wave up 1.
Its middle line is y = 1, not the x-axis.
middle line : y = 1
Careful!
The amplitude is from the middle line to the top, not from the bottom to the top.
Top to bottom is 4.
Careful: amplitude = 2, not 4
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.
0
π2
π
3π2
2π
A quarter of 2π is π/2.
a quarter of 2π = π2
The sin part follows the same pattern in every sine wave: 0, 1, 0, −1, 0.
0
1
0
−1
0
Each y is A times that, plus D: times 2, then plus 1.
1
3
1
−1
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 = π/2.
Put it in the equation.
x = π2: y = 2 × sin(π2) + 1
The inside is π/2, and sin π/2 is 1.
So y is 3, like in the table.
So the graph is right!
= 2 × sin π2 + 1 = 2 × 1 + 1 = 3
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.
▶ Watch it solved
The question
Question: graph y = 2 sin(2(x − π/4)) + 1.
y = 2 sin(2(x − π4)) + 1
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.
y = A sin(B(x − C)) + D
Here A is 2, the number in front.
B is 2, the number in front of x.
x minus C: C is π/4.
D is 1, the number added at the end.
A = 2
B = 2
C = π4
D = 1
2 Find the amplitude
The amplitude is how far the wave goes up and down from its middle line.
It is |A|: 2.
amplitude = |2| = 2
3 Find the period
One full wave of sin x is a full turn: 2π long.
B squeezes the wave: the period is 2π over B.
period = 2πB
Careful!
B squeezes the wave, so the period gets shorter.
We divide by B.
We do not multiply.
Careful: not period = 2π × 2 ✗
B is 2.
So the period is π: one full wave every π.
period = 2π2 = π
4 Find the shifts
C moves the wave sideways.
x − π/4 moves it right: the wave starts at x = π/4.
start : x = π4
D moves the wave up 1.
Its middle line is y = 1, not the x-axis.
middle line : y = 1
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/4.
π4
π2
3π4
π
5π4
A quarter of π is π/4.
a quarter of π = π4
The sin part follows the same pattern in every sine wave: 0, 1, 0, −1, 0.
0
1
0
−1
0
Each y is A times that, plus D: times 2, then plus 1.
1
3
1
−1
1
6 Draw the graph
Draw the grid.
The x-axis is in radians: a mark every π/4.
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 = π/2.
Put it in the equation.
x = π2: y = 2 × sin(2(π2 − π4)) + 1
The inside is π/2, and sin π/2 is 1.
So y is 3, like in the table.
So the graph is right!
= 2 × sin π2 + 1 = 2 × 1 + 1 = 3
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.
▶ Watch it solved
