Hey everyone! Trials are getting closer and closer, and you may be looking over your study notes (or these guides!) and saying to yourself, "How am I meant to remember the properties of all these things?" Look no further. This guide will summarise everything you need to know about log functions, quadratics, exponentials, trig functions, and for the extension students, inverse functions too. A quick read over this and you'll be ready to smash out your exams; over half of the questions have some relation to something in this section! This also means lots of overlap, so there will be concepts here covered in other guides (EG- Lots of calculus!). This guide does have lots to cover though, so I'll keep it brief where I can, but it will still be long. Consider it the directory guide

As always, remember to
register and pop any questions you had below, or in the 2/3 Unit Question Thread. There are awesome
notes available , which cover function theory in depth, and every other topic too!
To begin, some basic function theory, summarised in a few dot points.
- A function is an operator which takes an input (x) and gives an output (y)
- Each input must only give one output: This is what the vertical line test checks.
- The domain of the function is the range of x values which can be put into the function. The range is the range of outputs that you get from this domain.
- An
even function is a function where a positive input gives the same as the equivalent negative input, i.e.
=f\left(-x\right) )
Even functions are symmetrical about the y axis.
- An
odd function is a function where a positive input gives the same as the equivalent negative input, but opposite in sign. Ie:
=-f\left(x\right) )
Odd functions have a rotational symmetry of 180 degrees
As far as function theory goes, this is the extend of the knowledge you'll need. But make sure you know it! It may also help to take a peek at the guide I have written on The Number Plane, which goes into a few other little things like intercepts, asymptotes, etc, which are associated with graphing a function.
Below is a summary of the key functions you will see in a 2 Unit Exam (in their most basic forms), including domains, ranges, and other interesting behaviours.
CORRECTION: BOTH THE HYPERBOLIC AND TANGENT FUNCTIONS ARE DISCONTINUOUS The only function you really analyse in a great level of depth in 2 unit is the quadratic, so let's revise that a little bit. We remember that a quadratic is any function of the form:

It is a parabolic function. Let's prove some of the properties you should know.
Using calculus, we can show it has an axis at:

.
The axis occurs where the turning point occurs, so:

To find the vertex, we simply substitute in that x value to get the y value.
If a>0, then the quadratic has a positive concavity (parabola facing upwards)
If a<0, then the quadratic has a negative concavity (parabola facing downwards).
This can be proven easily with the second derivative.
You will likely know the quadratic formula from prior years, but you may not know that it is actually fairly simple to derive!
}^{2}=\frac{{b}^{2}-4ac}{4{a}^{2}}\\x+\frac{b}{2a}=\frac{\pm\sqrt{{b}^{2}-4ac}}{2a}\\x=\frac{-b\pm\sqrt{{b}^{2}-4ac}}{2a})
Cool huh! The part inside the square root is called the discriminant, and it determines how many solutions exist for the quadratic.
})
} )
} )
And finally, the roots of a quadratic can be linked with the formulae:

All of these together allow a comprehensive understanding of the quadratic, without the use of calculus. Let's look at a question which uses this:
This is a typical style question on roots of quadratics, with simple algebraic working.
=6\\2k=6\\k=3 )
Questions on vertexes, axes, and discriminants, are rarely asked individually. However, axes and vertex questions are integrated into questions on locus (covered in another guide), and the discriminant is
extremely useful in a variety of questions. It could pop up anywhere, it's an easy way to determine whether a solution exists or not. Combined with concavity, it can also prove that an expression is always positive or always negative.
The trig functions will be covered in a dedicated guide to trigonometry (this is the difficulty with these sorts of guides, choosing how to break them up is very difficult). However, be aware of their behaviour in a functional sense.
The exponential function appears mostly in differentiation and integration questions, due to it's nifty qualities and extensive applications in physics. Again, be aware that it never takes a negative value, and tends to infinity extremely quickly. Questions on exponentials are fairly simple, but remember the differentiation rules (the integration rules are the same in reverse, but they are also on the back of your exam!)
}=f'(x)e^{f(x)} )
Equally, you should remember the differentiation rules for logarithms!
 = \frac{1}{x} \quad\quad \frac{d}{dx} ln(f'(x)) = \frac{f'(x)}{f(x)} )
The logarithmic function also has some extra rules for you to remember. The Log Laws you should know, are listed below:
These rules are used often in calculus questions as well, but also, as standalone tests of the rules themselves.
The working below should be fairly simple.

This also highlights another cool fact,
logs and exponentials are exponential functions! You don't need to know this though.
Extension students have three other areas to deal with, polynomials, root estimation and inverse functions.
Spoiler
Polynomials are functions of the form

Some quick terminology; the leading term is the term with the highest power. The leading coefficient is the coefficient of this term. Degree of a polynomial is it's highest power. Remember, all powers in polynomials must be integers greater than or equal to zero.
The theorem to remember here is the remainder theorem (which is the general case of the factor theorem).
If p(x) is a polynomial, the remainder when the polynomial is divided by (x-a) is equal to p(a). Or, in other words:
=Q(x)(x-a)+P(a) )
where Q(x) is a polynomial of lower degree than the original.
This often links to advanced factorisation questions, or locating missing values:
Example (HSC 2014): The polynomial below has a factor x – 2. What is the value of k?
We apply the factor theorem to deduce that P(2), so:

You also have to know the more generalised cases of the root equations above.

The above is for a cubic equation. It is rare to get a quartic, but nothing prohibits it, so it might be worth some practice!
Next, estimation of roots. In the interest of brevity, I won't go through an example, but there are two methods you should know:
The Bisection Method involves taking two points either side of a root. One will have a positive sign, the other a negative. We can repeatedly bisect the interval between the two points, checking the sign as we do so, to reach more accurate estimations!
Newton's Method involves the derivative, and is to with the tangent of the curve at a chosen point. The formula for the more accurate root, given a point, is:
}{f'\left(x_1\right)})
Such questions are normally simple substitution style questions for easy marks. However, they will occasionally test you by asking which is a better method. The answer is, almost always, Newton's method. HOWEVER, if a turning point exists between the root and the test point, Newton's method fails.
That is asked repeatedly, so don't forget that! The final area of interest for extension students is inverse functions. An inverse function can be considered as a sort of reverse function, which takes the outputs of the original and gives back the original inputs. Now, questions on inverse functions specifically (not concerning the inverse trig functions) are rare, so again, in the interest of brevity, I will skip an example (I don't want you having to read through massive documents instead of studying, these are meant to be quick reads

). However, if you have a specific question, post it below, and I will happily walk everyone through it! Here are the main things you should know about inverse functions:
- Inverse functions exist only if each output can only be achieved through a single input. You likely know this as the horizontal line test, but technically, it means the function is what is called one-to-one.
- The inverse of a function can be found be reversing the positions of the y and the x, and rearranging.
- The inverse of a function reverses the domain and range. That is, the range becomes the domain and the domain becomes the range
- A function is symmetrical with it's inverse about the line y=x.
Again, detailed questions on inverse functions are rare. It usually sticks to finding an inverse and/or graphing it. But be sure to practice them, because lots of weird and wonderful things can be asked if BOSTES feels particularly nasty. Be sure to post any questions below and I will run through them for everyone to benefit from.
That's all guys! This guide is very much a bits and pieces sort of guide, linking a few different things together under the functions part of the course. Again, for more detail in terms of any of the areas, check out the other guides or the
notes available for 2/3 Unit Topics. Stay tuned for more guides, post questions, give me some feedback, all that good stuff

[/list]