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jamonwindeyer

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HSC Dysfunction: The Functions You'll See in 2/3 Unit
« on: June 21, 2015, 02:24:44 pm »
+8
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.



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:



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!



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.



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!)



Equally, you should remember the differentiation rules for logarithms!



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:



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:



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  8)

A GUIDE BY JAMON WINDEYER [/list]
« Last Edit: March 21, 2016, 08:51:22 pm by jamonwindeyer »

RuiAce

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #1 on: May 08, 2016, 05:42:25 am »
+3
I particularly struggle with quadratic equations, identifying and formulating the equations of graphs.
I also struggle in understanding how to dilate and reflect the graph correctly parallel to x and y axis  :'(
Can I please have some help?









anotherworld2b

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #2 on: May 08, 2016, 10:21:16 am »
0
Thank you for your explanation RuiAce. I've always been slightly confused whenever x and y axis is mentioned for transformations :D
I also apologize for asking such a vague question
- I actually wanted to ask how to change the graph of y= f(x) for example into:
y= f(0.5x)
y= 0.5f(x)
y= f(-3)
- What I meant by formulating equations was I actually have difficulty in how to identify and make an
equation given a graph of a function. eg. given a parabola, hyperbolic graph
and asked to write its equation.

- For quadratic functions I was confused on how:
y= ax^2 + bx + c
y= a(x-b)(x-c)
y= a(x-b)^2 + c
Can be used to find the y intercept, roots, turning point and ect
I hope this is clearer  :-[
Graphs and functions is my weakest point  so far

I appreciate all help and adive  :)

RuiAce

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #3 on: May 08, 2016, 11:09:59 am »
+1
Thank you for your explanation RuiAce. I've always been slightly confused whenever x and y axis is mentioned for transformations :D
I also apologize for asking such a vague question
- I actually wanted to ask how to change the graph of y= f(x) for example into:
y= f(0.5x)
y= 0.5f(x)
y= f(-3)
- What I meant by formulating equations was I actually have difficulty in how to identify and make an
equation given a graph of a function. eg. given a parabola, hyperbolic graph
and asked to write its equation.

- For quadratic functions I was confused on how:
y= ax^2 + bx + c
y= a(x-b)(x-c)
y= a(x-b)^2 + c
Can be used to find the y intercept, roots, turning point and ect
I hope this is clearer  :-[
Graphs and functions is my weakest point  so far

I appreciate all help and adive  :)















Side note: The Extension 1 student, on the other hand, can easily tell you the horizontal asymptote if you're given y=(x-B)/(x-D)
« Last Edit: May 08, 2016, 11:24:00 am by RuiAce »

anotherworld2b

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #4 on: May 08, 2016, 06:24:05 pm »
0
Thank you so much RuiAce for your help :D
For the sine, cos and tan graphs
I'm also having trouble in sketching the graph for questions like:
Sketch the graph of y= 3 cos (pi x) for 0 less than or equal to x less than or equal to 8
I can find the amplitude and period but I'm unsure on how to sketch a graph using this information.

Another example question is:
Sketch both of the following on a single pair of axes, with 0 less than or equal to x less than or equal to 360 degrees
a) y= 2 tan x    b) 2 tan(x+45 degrees)

Help is appreciated :)
« Last Edit: May 08, 2016, 06:31:02 pm by anotherworld2b »

jamonwindeyer

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #5 on: May 08, 2016, 09:38:37 pm »
+1
Thank you so much RuiAce for your help :D
For the sine, cos and tan graphs
I'm also having trouble in sketching the graph for questions like:
Sketch the graph of y= 3 cos (pi x) for 0 less than or equal to x less than or equal to 8
I can find the amplitude and period but I'm unsure on how to sketch a graph using this information.

Another example question is:
Sketch both of the following on a single pair of axes, with 0 less than or equal to x less than or equal to 360 degrees
a) y= 2 tan x    b) 2 tan(x+45 degrees)

Help is appreciated :)

Hey hey! Your best bet with these sorts of questions is to be able to sketch the basic trig functions off the top of your head (or at least derive them very quickly):



You then modify these graphs based on your given expression!

Take your example with the cosine curve:



This is a sketch of the cosine curve starting at cos0 and ending at cos8pi. You can verify that, for both of those points, y=3, if you need to. However, what is more effective is to simply say, okay: I know what my cosine curve looks like. The 3 out the front of the function is going to stretch the curve vertically, so that instead of going from -1 to 1 like the regular cosine curve does, you go from -3 to 3 instead. This is because, as the cosine term oscillates from -1 to 1 as normal, the 3 makes this range larger, tripling it. This is much the same ideas as y=2x being steeper than y=x.

In general, for the sine and cosine curves (using sine as an example):



This is a regular sine curve, stretched vertically to oscillate between A and -A. This is the amplitude. The 'B' term dictates the period/frequency of the curve (how far apart the peaks are from each other), and so stretches/compresses the curve horizontally. The 'C' term takes this curve, and shifts it to the left or right by 'C' units (depending on the sign). It is called a phase shift.

Try popping yourself in front of a graphing program for half an hour and playing with different numbers in this expression. Watch what the curve does to try and visualise what is happening. That is the absolute best way to go about it.

As for your second example, take the typical graph of y=tanx.



This curve will look almost identical, but stretched in the vertical direction for a similar reason to above. Now, when x=45 degrees, y will equal to instead of 1.



Notice that this is the same curve as before, with an additional term in the bracket. This is called a phase shift. What this means is that, whereas before when x=0, y=0; now we have x=0, y=2. Verify with calculator if you need to. What has happened here is that the curve has been shifted to the left by 45 degrees, now, the x-intercept is at x=-45.

I absolutely recommend playing around with a graphing program to see some of these effects for yourself! It is not too tricky once you get the hang of it, but getting there can be quite difficult. Let us know if you had any more examples we can explain for you, practice makes perfect for these sorts of questions!!  ;D

jakesilove

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #6 on: May 09, 2016, 10:33:00 am »
+1
Hey hey! Your best bet with these sorts of questions is to be able to sketch the basic trig functions off the top of your head (or at least derive them very quickly):



You then modify these graphs based on your given expression!

Take your example with the cosine curve:



This is a sketch of the cosine curve starting at cos0 and ending at cos8pi. You can verify that, for both of those points, y=3, if you need to. However, what is more effective is to simply say, okay: I know what my cosine curve looks like. The 3 out the front of the function is going to stretch the curve vertically, so that instead of going from -1 to 1 like the regular cosine curve does, you go from -3 to 3 instead. This is because, as the cosine term oscillates from -1 to 1 as normal, the 3 makes this range larger, tripling it. This is much the same ideas as y=2x being steeper than y=x.

In general, for the sine and cosine curves (using sine as an example):



This is a regular sine curve, stretched vertically to oscillate between A and -A. This is the amplitude. The 'B' term dictates the period/frequency of the curve (how far apart the peaks are from each other), and so stretches/compresses the curve horizontally. The 'C' term takes this curve, and shifts it to the left or right by 'C' units (depending on the sign). It is called a phase shift.

Try popping yourself in front of a graphing program for half an hour and playing with different numbers in this expression. Watch what the curve does to try and visualise what is happening. That is the absolute best way to go about it.

As for your second example, take the typical graph of y=tanx.



This curve will look almost identical, but stretched in the vertical direction for a similar reason to above. Now, when x=45 degrees, y will equal to instead of 1.



Notice that this is the same curve as before, with an additional term in the bracket. This is called a phase shift. What this means is that, whereas before when x=0, y=0; now we have x=0, y=2. Verify with calculator if you need to. What has happened here is that the curve has been shifted to the left by 45 degrees, now, the x-intercept is at x=-45.

I absolutely recommend playing around with a graphing program to see some of these effects for yourself! It is not too tricky once you get the hang of it, but getting there can be quite difficult. Let us know if you had any more examples we can explain for you, practice makes perfect for these sorts of questions!!  ;D

Hey!

Just to quickly add on to Jamon's fantastic summary, I would also recommend playing with a graphing calculator. Things like this can seem tricky, but basically once you've been forced to deal with graphs enough it will become almost second nature. I would recommend Desmos (the link for which you can find here). Put the graphs of the normal sine graph, a sine graph with a different amplitude, a sine graph with a different period and a sine graph with a different phase shift on top of each other. Just looking it that, it becomes pretty easy to see what each component does! Then, it is just about applying that to a question.

The last thing I would note is a 'what to do if you have no idea what to do' tip. I used to do this all the time, because for the life of me I couldn't distinguish between the Cosine and Sine graphs. Type it into a calculator. Say you are given the graph



That can seem pretty difficult, although if you understand how to shift graphs, you'll get it in no time. However, say you had no idea what to do, or just wanted to check that you got the right answer. Type that formula into your calculator, and just change 'x'. Draw yourself a set of axis with intuitive endpoints (ie. domain will be whatever they give you, range will be between the two amplitudes). Now, set x equal to zero, and plot the point that your calculator gives you. Then, set x equal to Pi/4, then Pi/2, etc. etc. for as many points as you want. This should help you get a general feel of what the graph looks like, and it should become fairly easy to sketch!

You shouldn't really be using this method to get the actual graph, although you can definitely do it to check that you're right. However, it's a good idea to keep this on tap in case your mind goes blank in an exam.

Jake
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jamonwindeyer

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #7 on: May 09, 2016, 11:30:45 am »
0
Hey!

Just to quickly add on to Jamon's fantastic summary, I would also recommend playing with a graphing calculator. Things like this can seem tricky, but basically once you've been forced to deal with graphs enough it will become almost second nature. I would recommend Desmos (the link for which you can find here). Put the graphs of the normal sine graph, a sine graph with a different amplitude, a sine graph with a different period and a sine graph with a different phase shift on top of each other. Just looking it that, it becomes pretty easy to see what each component does! Then, it is just about applying that to a question.

The last thing I would note is a 'what to do if you have no idea what to do' tip. I used to do this all the time, because for the life of me I couldn't distinguish between the Cosine and Sine graphs. Type it into a calculator. Say you are given the graph



That can seem pretty difficult, although if you understand how to shift graphs, you'll get it in no time. However, say you had no idea what to do, or just wanted to check that you got the right answer. Type that formula into your calculator, and just change 'x'. Draw yourself a set of axis with intuitive endpoints (ie. domain will be whatever they give you, range will be between the two amplitudes). Now, set x equal to zero, and plot the point that your calculator gives you. Then, set x equal to Pi/4, then Pi/2, etc. etc. for as many points as you want. This should help you get a general feel of what the graph looks like, and it should become fairly easy to sketch!

You shouldn't really be using this method to get the actual graph, although you can definitely do it to check that you're right. However, it's a good idea to keep this on tap in case your mind goes blank in an exam.

Jake

Good call Jake! Macs also come with a utility called Grapher, so that's another option if you have a Mac  ;D

RuiAce

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Re: HSC Dysfunction: The Functions You'll See in 2/3 Unit
« Reply #8 on: May 09, 2016, 12:17:47 pm »
+1
Good call Jake! Macs also come with a utility called Grapher, so that's another option if you have a Mac  ;D

Haha lol. One-up for this - saved me quite a few times whilst I was doing my uni homework.