Trigonometric Graphs 3
LI - To recognise and sketch the graphs of y = asinbxo + c and
y = acosbxo + c with different amplitudes and periods.
SC - I know the difference between a sine and cos graph
from the shape of the graph.
- I can write the equation of the graph in the form
y = asinbxo + c or y = acosbxo + c.
- I can sketch the graph of y = asinbxo + c or y = acosbxo + c.
Prior learning.
From the previous lessons, you should know that:
● y = sinxo passes through (0, 0) but y = cosxo passes through (0, 1).
● a tells us the value of the amplitude, the (vertical) distance from the x-axis
to the maximum.
● the period is the (horizontal) distance between identical points on
consecutive waves.
● b = 360 ÷ period and rearranging gives period = 360 ÷ b.
● You should be able to sketch and identify from it’s graph, the graphs of
y = asinbxo and y = acosbxo.
The graphs of y = asinbxo + c and y = acosbxo+ c.
y = 2cos4x + 3
y = 2cos4x
Examine the 2 waves above and draw a conclusion about what effect
adding 3 has had to the graph of y = 2cos4x.
Now examine this and draw a conclusion about what effect subtracting 1
has had on the graph of y = 5sin2x.
y = 5sin2x
y = 5sin2x - 1
You should have noticed that adding 3 has the effect of moving the graph
up 3 units.
Also, that subtracting 1 has the effect of moving the graph down 1 unit.
In other words, the y-coordinate has been altered but not the x-coordinate.
For y = asinbxo + c and y = acosbxo + c, the graph
moves up or down by c units.
If c is positive, the graph moves up c units.
If c is negative, the graph moves down c units.
State the equation of this graph in the form y = asinbx + c.
1.
2.
3.
4.
5.
Identify whether it is a sin or
cos graph.
Identify the value of the
amplitude.
Identify the value of b.
Identify the value of c.
Write equation in the form y
= ……
Sometimes it can be difficult to determine whether it is a sin or cos graph.
Sin graph.
Look at the y-axis. The graph is not at its
maximum or minimum point here.
Cos graph.
Look at the y-axis. The graph is at its
maximum or minimum point here.
Solution.
State the equation of this graph in the form y = asinbx + c.
1.
2.
3.
4.
Equation is y = 2sinx + 5
5.
Sin graph.
a = 2. The distance from
the horizontal centre to the
maximum is 2.
Or, the distance between
max and min is 4, so, the
amplitude is 4 ÷ 2 = 2.
Period = 360o. So, b = 360
÷ 360 = 1. Check, there is 1
complete wave in 360o✅.
c = 5. The graph has
moved up 5 units. [from (0,
0) to (0, 5) is 5 units]
Equation is y = 2sinx + 5
State the equation of this graph in the form y = asinbx + c.
1.
2.
3.
4.
5.
Identify whether it is a sin
or cos graph.
Identify the value of the
amplitude.
Identify the value of b.
Identify the value of c.
Write equation in the form
y = ……
1. Cos graph. The graph goes
Solution.
State equation of this graph in the form y = asinbx + c. through the y-axis at its
2.
3.
4.
5.
Equation is y = 3cos4x - 6
maximum
a = 3. The distance from the
horizontal centre to the
maximum is 3.
Or, the distance between
max and min is 6, so, the
amplitude is 6 ÷ 2 = 3.
Period = 90o. So, b = 360 ÷
90 = 4. Check, there are 4
complete waves in 360o✅.
c = -6. The graph has moved
down 6 units. [from (0, 0) to
(0, -6) is -6 units]
Equation is y = 3cos4x - 6
State equation of this graph in the form y = asinbx + c.
1.
2.
3.
4.
5.
Identify whether it is a sin or cos graph.
Identify the value of the amplitude.
Identify the value of b.
Identify the value of c.
Write equation in the form y = ……
Solution.
State equation of this graph in the form y = asinbx + c.
1.
2.
3.
4.
Sin graph as graph does not cut y-axis at max or min point.
a = -3. From horizontal centre to max is 3. However, this is negative due to shape of the graph.
period = 720o, so b = 360 ÷ 720 = 0.5. Check, there is 0.5 waves in 360o.✅
c = 2. The graph has moved up 2 units [from (0, 0) to (0, 2) is 2 units].
5.
Equation is y
= -3sin0.5x + 2.
Now complete these questions.
Similar to the previous assignment, the answers to these questions will be
posted in a couple of days. Please use these to check your work. There is no
need to submit your solutions to the questions above.
However, there is an assignment that you have to submit your answers
to. Good luck.