In my opinion, one of the more difficult topics in Extension 1 and 2 has got to be curve sketching. This is for a few reasons, but primarily because curve sketching takes a little bit of intuition. Knowledge of exactly what happens to a function when it is reflected, inverted, etc., is something which takes time to develop (even as a 2nd year university student I still get it wrong).
This resource contains a break-down of some of the curve sketching techniques required in MX1 and MX2, as well as a few tips and tricks to help you along the way! It is tailored for Extension 2, but the first section will definitely be applicable for MX1 students struggling with their 3 Unit curve sketches!
There are two types of curve sketching questions that you’ll see in your Extension exams; questions where you are given the function to sketch in an algebraic form, and questions where you are given the curve to sketch in a graphical form, then asked to modify it.
Without a doubt, the questions which give you the function in an algebraic form are much easier to handle. The process is more systematic. Mine goes something like this:
1.
Find Domain and Vertical Asymptotes: The first thing to consider, before any other train of thought, is the values which can be used as inputs for the function. Not considering this is a common trap for Extension 1 sketching questions. Make sure you have non-zero denominators and positive terms inside square roots, those are the two most common issues. You’ll also need to use a bit of intuition (or substitution) to determine the limiting values either side of this asymptote. Does it approach negative infinity, or positive infinity?
2.
Find Range and Horizontal Asymptotes: You have considered what can go in, so naturally the next step is to consider what can come out. Check that negative results are not excluded, or whether the function approaches a particular value for large values of x.
3.
Find Oblique Asymptotes: Sometimes, an asymptote may exist which is neither horizontal nor vertical. Instead, it is defined by a straight line equation of the form:
Oblique asymptotes are found by long division; we’ll go through an example in a little bit.
REMEMBER: While vertical asymptotes are super important restrictions, horizontal/oblique asymptotes are more like guidelines. It may be that a function touches its horizontal/oblique asymptote for low values of x, and then only approaches it for large values. 4.
x/y Intercepts: Often with all the complexities of asymptotes and calculus, we neglect the simple stuff, like the intercepts. Take the time to substitute x=0 and y=0 into your equation and see where it meets the axes.
5.
Calculus: This is where you do the calculus necessary for the question (or necessary for you to get a picture of what is happening). Turning points, inflexions, all can be found with differential calculus. Keep in mind however that, quite often, functions given in Extension 1 or 2 are quite complex. If the question doesn’t specify turning points, and you can get the curve without them, then avoid them. Differentiating them can waste time and be more prone to error.
6.
Plot Points: To finish, we draw the curve, possibly plotting extra points if we need a little more detail.
I used to remember the first 5 steps of this process as DROIC (Domain, Range, Obliques, Intercepts, Calculus). I’ve heard people use DROID, with the latter D simply meaning ‘Draw’ (they must not like Calculus). The point being, coming up with a method to remember this process is an extremely useful idea.
Let’s look at an example.
= \frac{x^3}{x^2-9} )
Step 1, find our vertical asymptotes. We cannot have a denominator of zero.
Now, let's look at what happens near these boundaries. Near x=3:
= -\infty \\ \lim_{x\to3^+}f(x)= \infty )
We can consider this by substituting in values VERY close to 3, either side of 3. More intuitively, however, we can deduce that substituting numbers just below 3 will yield a very small NEGATIVE denominator, and hence, the function skyrockets to extremely large negative values. Substituting numbers just above three will yield very small POSITIVE denominators, and thus, the function skyrockets to large positive numbers.
Step 2 and 3 involves finding our horizontal and oblique asymptotes. Both can be done with long division. Applying the typical process of long division, we find that we can express the function in a different way:
 = x+\frac{9x}{x^2-9} )
What we find here is interesting. As x becomes increasingly large, the fraction term becomes smaller and smaller, the denominator is growing much faster than the numerator. Thus, for large values of x, the value of y is actually approaching the value of x, always just a tiny little bit bigger. Thus, we have an oblique asymptote at y=x.
Of course, this can just be read from the above form. Note also that a horizontal asymptote would be found in the same way. For example, applying long division to the function below, we can find a horizontal asymptote of y=2. The result is, in fact, related to the degree of the numerator vs the degree of the denominator. When the degrees are equal, the asymptote is horizontal, when the degree of the numerator is higher, it is an oblique asymptote. These are things to remember, though I always found just doing the long division easier. Less likely to trick you. = \frac{2x^2+5}{x^2-25} = 2+\frac{55}{x^2-25} )
Okay, back to the original function. We have asymptotes now. We can fairly easy get intercepts by substitution, and it is found that the only intercept present is actually the origin.
But wait! Isn't the line y=x an asymptote? Well, yes, but it only has this property for large values of x. The origin is a perfectly valid inclusion for the curve. Watch out for trick cases like this, they can be confusing!
We don't need any Calculus to get this done, but if it helps, feel free:
=\frac{x^3}{x^2-9} \\ f'(x) = \frac{3x^4-27x^2-2x^4}{(x^2-9)^2} = \frac{x^4-27x^2}{(x^2-9)^2} )
You can solve for critical points at 0, and plus minus 3 root 3. This may help you get a better picture of what is happening.
So we have obtained:
- Formula for the asymptotes and behaviour around these asymptotes
- Intercept at the origin
- (Optional) Critical Points
Putting all this together, we obtain the following graph. The function is in green. We can see the critical points quite clearly, and I've added the oblique asymptote in black. The vertical asymptotes, while not shown, are clearly at positions x = +-3 as determined.
This is just one example with one combination of conditions. There are many, so the best way to prepare is to practice. Use this method (or come up with your own) and try it on as many past questions as you can. If you run into a question that stumps you,
feel free to post it below!Right, and that covers the cases where you are given the function.
What about questions where you aren't?A very typical question in MX2 will give you the sketch of a function, then ask you to sketch/identify the sketch of that function, modified in some way. Some examples are below:
 \quad \sqrt{f(x)} \quad \frac{1}{f(x)} \text{...} )
Some people find these questions easier. They often require little to no written working, which is nice. However, they rely quite heavily on intuition, and a deep understanding of functions. They will try and trick you.
Again, there is no way to prepare you for every possible question. However, the questions do generally contain one of two elements, or sometimes both. The input may be modified, or the output may be modified.
Input modifications look something like this:
 \quad f(-x) \quad \text{etc...} )
Here are some tricks for dealing with some of the common ones:
\text{ are quite simple to sketch. You simply reflect the curve about the y axis.} )
 \text{ are a little tougher. Some easy tricks - The function is undefined for negative x values.}\\\text{ The new function will have the same value as the old one for x=0 and x=1} )
\text{, simply take the function as defined for positive x values, and reflect it for negative x values.} )
Output modifications involve modifications to the function value itself. To deal with some common ones:
} \text{, means that all negative function values are removed. For positive function values, a few things } \\\\ \text{can happen. For f(x) between 0 and 1, the new function is greater. For f(x) larger than 1, } \\\\ \text{the new function is smaller. For 0 and 1, they are equal.} )
|\text{, simply takes the negative function values and makes them positive, } \\\\\text{effectively reflecting negative sections about the x-axis.} )
} \text{, is interesting. Wherever f(x) equals zero, we will have vertical asymptotes.} \\\\ \text{As f(x) gets big, it's reciprocal will get small, and vice versa. There will be no sign changes either.} )
These are ways to handle just a few of the common situations. There are HEAPS of these, combined in various ways, and it would be totally unreasonable to memorise circumstances for all of them. Much more effective is developing an
intuition for these questions. Take the modification and consider the sign and magnitude at the result, will certain values now be excluded? Will this new function now be huge or tiny? Positive or negative? Some careful consideration will quite often reveal the answer, especially with practice.
Even more tips and tricks courtesy of RUIACE! These are more complex, less obvious situations, definitely worth remembering if you have the time, or if not, definitely worth understanding! 
.
When taking powers:
]}^{2}, {[f(x)]}^{4}, \dots \rightarrow x\text{-intercepts remain as }x\text{-intercepts, but they all become stationary points})
]^3, \dots\rightarrow x\text{-intercepts also remain, but they all become horizontal points of inflexion.}<br /><br />[tex]\text{Proof: Use the chain rule to find }\frac{d}{dx} {[f(x)]}^{n}\text{ for some integer n)})
}\right)\text{ also reverse the nature of stationary points - your local min become local max and etc.})
____________________________
 to find your slant asymptote})
My advice would be to start simple. Work from the simple things, then start combining them into more complex functions. Slowly but surely, you will start to instinctively know what the curve should look like!
Hopefully this guide helps you with these sorts of questions! MX2 curve sketching can be a little nasty, so by all means, feel free to post any questions below for an extra helping hand!
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