T2W2 BL

You are to complete this BL by 2/4 11:59pm.

In this BL session, you will be covering:

Chapter 3A Graphing Techniques (Section 3: Curves of Rational Functions)

Use the following learning activities to master the key concepts. Rem to fill in your lecture notes as you go along!

You should take 1 hour to complete this lesson.

Quick Recall: Last Lesson

Last lesson, we learnt what are these key features of graphs and how to find them:

Axial Intercepts
Stationary Points
Asymptotes (In particular, we learnt how to identify the Asymptotes of Rational Functions)

In this session,

We will be looking at how to sketch the curves of rational functions.

Section 3 Curves of Rational Functions

In general, before we sketch the curves of rational functions, we typically

(1) Use the GC to observe the BEHAVIOUR/SHAPE of the graph

(2) Find the key features of the graph either by algebraic working or using the GC

Considering Key Features of Curves of Rational Functions

Step 1: Find where the graph crosses or intersects the axes.

Step 2: Identify the asymptotes of the graph.

Step 3: Identify any stationary points (local maxima, minima or stationary points of inflection).

Try using your GC to see what the graph looks like! Follow the steps in the video if you are unsure!

What do you observe?

Notice that the graph in the GC does not show explicitly the asymptotes of the graph even though it shows the asymptotic behaviour of the graph.
From the graph of the GC, we can possibly expect a horizontal asymptote and a vertical asymptote!

Let's proceed to countercheck algebraically!

2. Attempt to fill in the blanks in your lectures notes yourself first!

Read and check your solutions below!

We can also use our GC to find axial intercepts!

Watch this video if you are not sure how to do so!

Try using your GC to see what the graph looks like! Follow the steps in the video if you are unsure!

What do you observe?

From the graph on the GC, we can possibly expect a horizontal asymptote and a vertical asymptote!
(Try yourself!) Find the axial intercepts using your GC.

Let's proceed to countercheck algebraically!

2. Attempt to fill in the blanks in your lectures notes yourself first!

Read and check your solutions below!

Try using your GC to see what the graph looks like! Follow the steps in the video if you are unsure!

What do you observe?

From the GC, we can possibly expect a vertical asymptote and an oblique asymptote.
(Try yourself!) Find the axial intercepts using your GC.
Also, unlike the previous 2 examples, we now observe that there are stationary points!

Let's proceed to countercheck algebraically!

2. Attempt to fill in the blanks in your lectures notes yourself first!

Read and check your solutions below!

We can also use our GC to find stationary points!

Watch this video if you are not sure how to do so!

What do you observe?

The stationary points that the GC generates will be in non-exact form

Additional Note

Typically if a question does not ask for the exact intercepts and stationary points, it is acceptable to use your GC to find them and label the coordinates of these points in 3 s.f. (There is no need to find these points algebraically)

3. Sketching the Graph

Watch the following video and follow the steps to sketch your graph!

Turn on audio to listen to the lecturer explain :)

In general, when you sketch a graph:

Start with sketching the asymptotes of the graph as dotted lines (using ruler!)
Mark out the intercepts and stationary points (where applicable)
Sketch in the curve
Label the equation of the graph

Asymptotic Behaviour

Where the curve tends to the asymptotes, you are to ensure that the curve is getting CLOSER and CLOSER to the asymptotes but NOT TOUCHING the asymptotes.

What is wrong with the red curve on the left?

1. The curve is too far away, and almost running "parallel" to the asymptote

2. The curve is not tending to the asymptote but veering away from it

3. The curve stops abruptly

4. The curve is touching the asymptote

These are INCORRECT ways of representing asymptotic behaviour.

Asymptotic Behaviour

Where the curve tends to the asymptotes, you are to ensure that the curve is getting CLOSER and CLOSER to the asymptotes but NOT TOUCHING the asymptotes.

Further Exploration

Try out each question yourself first before checking your solutions below!