ATAR Notes: Forum

VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: pi on February 16, 2012, 04:51:17 pm

Title: [GUIDE] Techniques for Sketching Nice-Looking Graphs
Post by: pi on February 16, 2012, 04:51:17 pm: Techniques for Sketching Nice-Looking Graphs (Guide)

So, throughout my time being a spesh student, I saw some pretty horrendous looking graphs. Personally, I think that drawing nice looking graphs using solid techniques and a pencil+ruler (this is important!) can do any of the following:
1) Speed up the time you spend on a question
2) Makes it easier for not only the examiner to find things, but also yourself
3) Impress an examiner (this is important!)

For me, I found that visualising points was equally as important as calculating them, and in fact, the simple visualisation of certain points can make a complicated graph very easy to sketch. Just remember, "sketch" does not mean messy!

So here's my quick guide on how I like to go about things :)

Implied domains of certain scenarios

#1
$y = \frac{f(x)}{g(x)}$
Take $g(x)=0$ and solve for $x= x_1, x_2, ...$
Domain = $R$ \{ $x_1, x_2, ...$ }

#2
$y = \sqrt{f(x)}$
Take $f(x) \ge 0$ and solve (use a "quick-sketch" if needed)
Domain = solution of inequation

#3
$y= \frac{f(x)}{\sqrt{g(x)}}$
Take $g(x)>0$ and solve (use a "quick-sketch if needed)
Domain = solution of inequation

#4
$y=\log_n{(f(x))}$
Take $f(x)>0$ and solve
Domain = solution

Sketching a nice looking graph (addition of ordinates)
1. Recognise the case: $f(x)=y=ax^m+bx^{-n}+c=g(x)+h(x)$ , $m \ or \ n=1 \ or \ 2$
   (i) Straight line and hyperbola
   (ii) Parabola and hyperbola
   (iii) Straight line and truncus
   (iv) Parabola and truncus

2. Find Domain and Range, $Domf=Domg \cap Domh$

3. Find any vertical asymptotes:
⇒ $x=a$ for $f(a)$ is undefined

4. Find any oblique or curved asymptotes:
⇒ Resolve $\lim_{x \rightarrow \infty}{y}$

5. Find critical points:
   (i) $y$ -int, let $x=0$
   (ii) $x$ -int, let $y=0$
   (iii) Stationary points, let $\frac{dy}{dx}=0$ and solve for $x$ and $f(x)$
   (iv) Any crossing of the horizontal asymptote, let $y=a$ and solve equation for $x$
   (v) Endpoints (if any)

6. Do a light dotted sketch of both $g(x)$ and $h(x)$ .

7. Find key points (used to aid graphing):
   (i) Zeroes of $g(x)$ and $h(x)$ . The $y$ -co-ordinate is on the other curve
   (ii) Cancelling points, the is $x$ -int (don’t solve for these points, do this visually), this should match your above calculation
   (iii) Visually use $y$ -ints of $g(x)$ and $h(x)$ to find the $y$ -int of $f(x)$ , this should match your calculation
   (iv) Intersections of $g(x)$ and $h(x)$ . The $y$ -co-ordinate is double of this.

8. Look left/right of each key-point, realising the behaviour of the curve

9. Sketch, rub-out any unnecessary dotted line graphs

10. Label the graph, axes, all asymptotes with their equations (as Asym $y=...$ or Asym $x=...$ ) and all critical points in co-ordinate form

Sketching a nice looking graph (reciprocation)*
1. Recognise curve as $y=\frac{1}{f(x)}$ (or manipulate mentally to see this)

2. Draw a light dotted sketch of $f(x)$

3. Horizontal asymptote is $y=0$

4. Draw vertical asymptotes through $x$ -ints of $f(x)$

5. Find key points:
   (i) $f(x)=\pm 1$ , these points will also be on $y$
   (ii) Stationary points, let $\frac{dy}{dx}=0$ and solve for $x$ and $f(x)$
   (iii) Endpoints (if any)

6. If $f(x) \rightarrow \infty$ , $\frac{1}{f(x)} \rightarrow 0$ , if $f(x) \rightarrow 0$ , $\frac{1}{f(x)} \rightarrow \infty$

7. Sketch, rub-out any unnecessary dotted line graphs

8. Label the graph, axes, all asymptotes with their equations (as Asym $y=...$ or Asym $x=...$ ) and all critical points in co-ordinate form

*If $y$ has been translated vertically, then there may be $x$ -ints and the horizontal asymptote will also change. These need to found if this is the case.

What examiners like to look for
- General shape
- Appropriate and realistic scaling used
- Correct $x$ -ints if they exist in co-ordinate form
- Correct $y$ -ints if they exist in co-ordinate form
- Correct local max/mins if they exist in co-ordinate form
- Correct end-points if they exist in co-ordinate form
- Correct and labelled asymptotes if they exist
What they like in addition to above:
- Labelled axes
- Labelled graph
- Domain and range given
- Straight lines done with a ruler
- No deviations away from an asymptote
- Clear labels, no smudging, good presentation

Example (for partial fractions - a twist on 'reciprocations')
From my notes, so it's not to the standard I would have done in a SAC or exam, but its alright and shows the working too :) Apologies for my crappy hand-writing too :D
(http://i40.tinypic.com/555iro.jpg)
Notes:
- I have left the "dotted-lines" of the various "parts" to the full graph, this is because this was done for notes purposes. You should erase these in SACs/exams.
- I have jumped to the second line of working, I used a CAS to save time and get the problem on one page :P
- Against my tips, I have labelled my asymptotes as only " $x=-1$ " instead of "Asym $x=-1$ "

N.B.
- For any asymptotes that may appear to lie on an axis, draw it in a coloured pen (preferably blue) or just above the axis.
- You may or may not need to prove a stationary point. If you do, I'd suggest either the use of second derivatives or a gradient-sign table. Check the question to make sure.
- After some practise, you may feel comfortable with skipping some of the steps listed above, nut I find that the above list is a very good start for "beginners".

Hope this helped, post any queries/suggestions/errors in this thread, good luck! :)
Title: Re: Techniques for Sketching Nice-Looking Graphs (Guide)
Post by: Lasercookie on February 16, 2012, 05:06:23 pm: If it's not any trouble, would you be able to scan in a couple of examples of your graphs/sketches? I think being able to see that will help give me a better idea of what level of neatness I should be aiming for.
Title: Re: Techniques for Sketching Nice-Looking Graphs (Guide)
Post by: pi on February 16, 2012, 05:09:09 pm: Quote from: laseredd on February 16, 2012, 05:06:23 pm
If it's not any trouble, would you be able to scan in a couple of examples of your graphs/sketches? I think being able to see that will help give me a better idea of what level of neatness I should be aiming for.

Will do :)

Added a new section at the end too :)
Title: Re: Techniques for Sketching Nice-Looking Graphs (Guide)
Post by: pi on February 16, 2012, 05:27:55 pm: Added an example, its not perfect (as per my techniques), but its the best I could find atm :)

lol, you can see my trig notes a bit too :D