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2 - Curve Sketching 1 (Lecture)

The document provides an overview of curve sketching techniques, focusing on asymptotes, including vertical, horizontal, and slant asymptotes, and their implications for rational functions. It outlines various cases for sketching curves based on the types of rational functions, including those with linear and quadratic numerators and denominators. Additionally, it includes practice questions for applying these concepts.

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lalamachola
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109 views5 pages

2 - Curve Sketching 1 (Lecture)

The document provides an overview of curve sketching techniques, focusing on asymptotes, including vertical, horizontal, and slant asymptotes, and their implications for rational functions. It outlines various cases for sketching curves based on the types of rational functions, including those with linear and quadratic numerators and denominators. Additionally, it includes practice questions for applying these concepts.

Uploaded by

lalamachola
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Curve Sketching 1 (lecture)

• An asymptote of a curve is a line or a curve whose distance


from the curve approaches zero as 𝑥 or 𝑦 or both
coordinates tend to infinity.
It shows the behaviour of the curve towards infinity.
• A curve can never cross a vertical asymptote.
• A curve can cross a horizontal or slant asymptote only if the
rational function consists of non-linear polynomials in the
numerator and denominator.
• On either side of a vertical asymptote the graph extends in
opposite directions, if one side extends to +∞ , then the
other side extends to −∞ . And conversely true.

Case1: Rational function with linear denominator


1
o Start with the basic 𝑦 = graph and transform it into the
𝑥
one required.
o 𝑦 − intercept.
𝐴𝑙𝑡:
o Vertical asymptote with 𝑥 −axis as horizontal asymptote.
o 𝑦 −intercept

Q1. Sketch the following,


2
a. 𝑦 =
𝑥+1
3
b. 𝑦 =
2−𝑥

Sherry
Case2: Rational function with linear numerator and
denominator
o Vertical and horizontal asymptotes.
o 𝑥 − 𝑎𝑛𝑑 𝑦 − intercepts.

Q2. Sketch the following,


4𝑥−8
a. 𝑦 =
𝑥+3
2𝑥−6
b. 𝑦 =
𝑥−5

PRACTISE WORKSHEET 1

Case3: Rational function with quadratic numerator and linear


denominator
o Vertical asymptote.
o Break the improper fraction to obtain an oblique asymptote.
o Turning points, if any.
o 𝑥 − 𝑎𝑛𝑑 𝑦 − intercepts, if any.

Q3. Sketch the following,


𝑥 2 −3𝑥+3
a. 𝑦 =
𝑥−2
𝑥 2 −9
b. 𝑦 =
1−𝑥

Sherry
Q4. (J14/12/Q12 OR)

PRACTISE WORKSHEET 2

Case4: Rational function with quadratic denominator


• The function may cross the horizontal asymptote.
• Generally, the following are to be found,
o Horizontal asymptote.
o Horizontal asymptote intercept, if any.
o Vertical asymptotes, if any.
o 𝑥 − 𝑎𝑛𝑑 𝑦 − intercepts.
o Break into proper fraction and determine the turning
point(s), if any.
• When the denominator has a repeating factor (repeating
root) we have what we call coinciding asymptotes.
o Graph on both sides of coinciding asymptotes extends
in the same direction either towards +∞ or −∞.

Sherry
1. Quadratic denominator with two vertical asymptotes.

Q5. Sketch the following,


1
a. 𝑦 =
(𝑥−1)(𝑥−2)
𝑥
b. 𝑦 =
𝑥 2 −2𝑥−3
𝑥 2 −7𝑥+14
c. 𝑦 =
𝑥 2 −4𝑥+3

Q6. (J14/13/Q11 EITHER)

2. Quadratic denominator with coinciding vertical asymptotes.


𝑥
Q7. Sketch 𝑦 =
(𝑥+2)2

Q8. (J13/12/Q10)

Sherry
3. Quadratic denominator resulting in no vertical asymptote.
• As there are no values for which the function tends to
infinity, the curve is therefore continuous (no breaks).

Q9. Sketch the following,


2𝑥 2 +5𝑥+3
a. 𝑦 =
4𝑥 2 +5𝑥+3
𝑥 2 −𝑥+2
b. 𝑦 =
𝑥 2 −𝑥+1
𝑥 2 −𝑥−1
c. 𝑦 = 1
𝑥 2 +𝑥+1
2

Q10. (N11/13/Q10)

PRACTISE WORKSHEET 3

PRACTISE PASTPAPERS

Sherry

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