 Hong Kong
Stage 4 - Stage 5

Lesson

The general form of the equation of a cosine curve is

$f\left(x\right)=a\cos\left(bx-c\right)+d$f(x)=acos(bxc)+d

From the lessons on key features and transformations, we discovered that

 $a$a amplitude, the maximum deviation of the graph from the central line $\frac{2\pi}{b}$2πb​ period, the length needed for the curve to travel one full cycle $\frac{c}{b}$cb​ phase shift, the horizontal translation of the curve $d$d vertical translation, the shift of the central line (and hence the entire curve) up or down.

In order to graph sine curves there are a number of different approaches you can take.  Which one you choose may depend on your own preference or the question you are given.  These are the same approaches that we looked at for sketching sine curves, but we will look at them again now using different functions.

### Approach 1

Walk through the transformations and change the stem graph $y=\cos x$y=cosx accordingly.

#### Example

##### Example 1

Graph $y=-1.5\cos\left(2x-\frac{\pi}{2}\right)-1$y=1.5cos(2xπ2)1

Start with a sketch of $y=\cos x$y=cosx Reflection - the first step is to check for a reflection, as the value of $a$a is less than $0$0, there is a reflection.  So now reflect the graph of $y=\cos x$y=cosx over the central line, this the graph of $y=-\cos x$y=cosx Apply vertical translation - move the graph down  units, this in now the graph of $y=-\cos x-1$y=cosx1 Increase the amplitude - the amplitude of this graph is $1.5$1.5 units, so we dilate the graph.  Move the maximum and minimum out an extra half unit.  This is now the graph of $y=-1.5\cos x-1$y=1.5cosx1 Period - the period of the function has been changed from $2\pi$2π to $\frac{2\pi}{2}=\pi$2π2=π so this is a horizontal dilation.   Shrink in the graph, keeping the starting point the same.  This is now the graph of $y=-1.5\cos\left(2x\right)-1$y=1.5cos(2x)1. Phase shift - the last transformation we need to consider is the phase shift.  We calculate it using $\frac{c}{b}=\frac{\frac{\pi}{2}}{2}=\frac{\pi}{4}$cb=π22=π4.  This means the graph is shifted to the right by $\frac{\pi}{4}$π4.  This is now the graph of $y=-1.5\cos\left(2x-\frac{\pi}{2}\right)-1$y=1.5cos(2xπ2)1 ### APPROACH 2

Step out all the important components and create a dot-to-dot style map of the function.

#### Example

##### EXAMPLE 2

Graph $-1.5\cos\left(2x-\frac{\pi}{2}\right)-1$1.5cos(2xπ2)1

Step 1 - identify the transformations from the graph by identifying the following

 $a$a amplitude $1.5$1.5 sign of $a$a reflection there is a reflection as $a$a is negative $\frac{2\pi}{b}$2πb​ period $\frac{2\pi}{2}=\pi$2π2​=π $\frac{c}{b}$cb​ phase shift $\frac{\frac{\pi}{2}}{2}=\frac{\pi}{4}$π2​2​=π4​ $d$d vertical translation $-1$−1

Step 2 - start by sketching the central line (indicated by the vertical translation) This is the line $y=-1$y=1 Step 3 - mark on the maximum and minimum by measuring the amplitude above and below the central line.  The amplitude is $1.5$1.5, so mark on the lines $y=-1+1.5=0.5$y=1+1.5=0.5, and $y=-1-1.5=-2.5$y=11.5=2.5 Step 4 - The phase shift is $\frac{\pi}{4}$π4, so this is a horizontal translation of $\frac{\pi}{4}$π4 to the right.  Mark a line at $x=\frac{\pi}{4}$x=π4.  This is where the function will begin. Step 5 - mark out the distance of the full period.  At this stage mark out half way and quarter way marks, this will help us sketch the curve. The full period needs to be marked out from the starting position.  The period is $\pi$π, so we will need lines at $\frac{\pi}{4}+\pi$π4+π (the end point) $\frac{\pi}{4}+\frac{\pi}{2}$π4+π2 (the halfway point) and then some guidelines halfway between these. Step 6 -The reflection changes the initial starting direction.  A cosine curve normally starts at its maximum and decreases, but a reflection means the cosine curve will start at the minimum and increase. I always mark the starting position with a dot and a small arrow indicating direction. Step 7 - Create some dots marking the ending position of the cycle and the centre, maximum ect. use the quarter lines we sketched earlier. Step 8 - Sketch the curve lightly at first, joining our preparatory dots together. Developing the skills for smooth curve drawing takes practice so don't get disheartened.  When you have a nice curve, draw it in. Some people prefer step by step constructions, some prefer the fluid changes of transformations, others develop their own order and approach to sketching cosine functions.  Regardless of your approach they will all need to use the specific features of the cosine curve.

#### More Worked Examples

##### QUESTION 1

Which of the following is the graph of $y=\cos x+3$y=cosx+3?

A

B

C

D

##### QUESTION 2

Consider the function $y=\cos\frac{4}{3}x$y=cos43x.

1. Identify the amplitude of the function.

3. Graph the function.

##### QUESTION 3

Consider the function $y=2\cos\left(x-\frac{\pi}{2}\right)+3$y=2cos(xπ2)+3.

2. Determine the amplitude of the function.

3. Determine the maximum value of the function.

4. Determine the minimum value of the function.

5. Graph the function.