A-level Further Maths Year 12 · Topic 4
Curves and graphs
Further Maths brings a whole family of new curves.
We will sketch rational functions with asymptotes and meet the parabola, ellipse and hyperbola.
Then we draw curves in polar coordinates and meet the hyperbolic functions.
- 5 subtopics
- 121 questions
- Free: 4.1 notes and 10 questions
In this topic
- 4.1Rational functions: asymptotes and curve sketchingFree below · 42 questions
- 4.2Conic sections: the parabola, ellipse and hyperbolaOpen in Full Marks · 14 questions
- 4.3Polar coordinates: converting and sketching polar curvesOpen in Full Marks · 11 questions
- 4.4Hyperbolic functions: definitions, graphs and identitiesOpen in Full Marks · 16 questions
- 4.5Inequalities involving rational functions and the modulusOpen in Full Marks · 38 questions
4.1 · Free notes
Rational functions: asymptotes and curve sketching
Key points
- A rational function is a fraction of polynomials, such as y = (2x + 1)/(x − 3).
- Vertical asymptotes occur where the denominator is zero (and the numerator is not): x = 3 above.
- Horizontal asymptotes describe the behaviour as x → ±∞: for equal degrees, y → ratio of leading coefficients (y = 2 above); if the denominator's degree is higher, y → 0. An oblique asymptote occurs when the numerator's degree is one more than the denominator's (divide to find it).
- To sketch: find intercepts (x = 0 and y = 0), asymptotes, the behaviour either side of vertical asymptotes, and any stationary points.
- The range of y = (ax² + bx + c)/(dx² + ex + f) can be found by rearranging into a quadratic in x and requiring the discriminant ≥ 0.
In more detail
- Worked example: y = (x² + 2)/(x − 1) = x + 1 + 3/(x − 1): vertical asymptote x = 1 and oblique asymptote y = x + 1.
- Behaviour check: test a value just either side of a vertical asymptote to see whether y → +∞ or −∞.
- Exam tip: label every asymptote with its equation and every intercept with coordinates on your sketch.
Practice
Rational functions: asymptotes and curve sketching: 10 questions
Pick an answer to check it, or open the answer for the explanation.
1. What is the vertical asymptote of y = (2x + 1)/(x − 3)?
- x = −3
- x = −1/2
- x = 3
- y = 2
Show answer
C: x = 3. The denominator is zero at x = 3.
2. What is the horizontal asymptote of y = (2x + 1)/(x − 3)?
- y = 2
- y = 0
- y = 3
- y = 1/3
Show answer
A: y = 2. Ratio of leading coefficients.
3. What is the horizontal asymptote of y = 5/(x² + 1)?
- y = 5
- y = 1
- None
- y = 0
Show answer
D: y = 0. The denominator grows faster.
4. Where does y = (x − 4)/(x + 2) cross the x-axis?
- (−2, 0)
- (4, 0)
- (0, −2)
- (0, 4)
Show answer
B: (4, 0). Numerator zero at x = 4.
5. Where does y = (x − 4)/(x + 2) cross the y-axis?
- (0, 4)
- (0, 2)
- (0, −2)
- (0, −4)
Show answer
C: (0, −2). −4/2 = −2.
6. Which function has an oblique asymptote?
- y = (x² + 1)/x
- y = (x + 1)/x
- y = 1/x²
- y = x/(x² + 1)
Show answer
A: y = (x² + 1)/x. Numerator degree one more: y = x + 1/x, asymptote y = x.
7. What are the vertical asymptotes of y = 1/((x − 1)(x + 2))?
- x = −1 and x = 2
- y = 0
- x = 1 only
- x = 1 and x = −2
Show answer
D: x = 1 and x = −2. Denominator zero.
8. As x → ∞, what does y = (3x² − x)/(x² + 4) approach?
- 0
- 3
- ∞
- −1/4
Show answer
B: 3. Leading coefficients 3/1.
9. How can you find the range of a rational function of two quadratics?
- Differentiate twice
- Find the y-intercept only
- Rearrange into a quadratic in x and use discriminant ≥ 0
- Use Pascal's triangle
Show answer
C: Rearrange into a quadratic in x and use discriminant ≥ 0. Real x requires a non-negative discriminant.
10. What does the curve do near a vertical asymptote?
- Tends to +∞ or −∞
- Crosses it
- Stops
- Becomes horizontal
Show answer
A: Tends to +∞ or −∞. y grows without bound.
Keep going in Full Marks
Full Marks has 42 questions on Rational functions: asymptotes and curve sketching (32 more than here), set out in practice levels. You also get notes and 3D models for all 5 subtopics of Curves and graphs, plus games, timed exams and past papers.
On your exam board
- AQATopic 3: Further algebra and functions, Topic 6: Polar coordinates, Topic 7: Hyperbolic functions
- EdexcelTopic 8: Option: Further Pure 1
- CambridgeTopic 2: Rational functions and graphs, Topic 5: Polar coordinates
Not in the OCR A and OCR B specifications.