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
}{g(x)})
Take
=0)
and solve for
Domain =
\{
}
#2
})
Take
 \ge 0)
and solve (use a "quick-sketch" if needed)
Domain = solution of inequation
#3
}{\sqrt{g(x)}})
Take
>0)
and solve (use a "quick-sketch if needed)
Domain = solution of inequation
#4
)})
Take
>0)
and solve
Domain = solution
Sketching a nice looking graph (addition of ordinates)1. Recognise the case:
=y=ax^m+bx^{-n}+c=g(x)+h(x))
,
(i) Straight line and hyperbola
(ii) Parabola and hyperbola
(iii) Straight line and truncus
(iv) Parabola and truncus
2. Find Domain and Range,
3. Find any vertical asymptotes:
⇒
for
)
is undefined
4. Find any oblique or curved asymptotes:
⇒ Resolve
5. Find critical points:
(i)
-int, let
(ii)
-int, let
(iii) Stationary points, let
and solve for
and
)
(iv) Any crossing of the horizontal asymptote, let
and solve equation for
(v) Endpoints (if any)
6. Do a light dotted sketch of both
)
and
)
.
7. Find key points (used to aid graphing):
(i) Zeroes of
and
. The
-co-ordinate is on the other curve
(ii) Cancelling points, the is
-int (don’t solve for these points, do this visually), this should match your above calculation
(iii) Visually use
-ints of
and
to find the
-int of
)
, this should match your calculation
(iv) Intersections of
and
. The
-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
or Asym
) and all critical points in co-ordinate form
Sketching a nice looking graph (reciprocation)*
1. Recognise curve as
})
(or manipulate mentally to see this)
2. Draw a light dotted sketch of
3. Horizontal asymptote is
4. Draw vertical asymptotes through
-ints of
5. Find key points:
(i)
=\pm 1)
, these points will also be on
(ii) Stationary points, let
and solve for
and
(iii) Endpoints (if any)
6. If
 \rightarrow \infty)
,
} \rightarrow 0)
, if
 \rightarrow 0)
,
} \rightarrow \infty)
7. Sketch, rub-out any unnecessary dotted line graphs
8. Label the graph, axes, all asymptotes with their equations (as Asym
or Asym
) and all critical points in co-ordinate form
*If
has been translated vertically, then there may be
-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
-ints if they exist in co-ordinate form
- Correct
-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
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
- Against my tips, I have labelled my asymptotes as only "
" instead of "Asym
"
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!
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