Graphs of rational functions and asymptotes
A rational function is one polynomial divided by another, like y = (x + 1)/(x − 2).
An asymptote is a line the curve gets closer and closer to but does not reach (far away).
- Vertical asymptote: where the bottom is 0 (and the top is not). Here x = 2.
- Horizontal asymptote for linear/linear: y = (top x-coefficient)/(bottom x-coefficient). Here y = 1/1 = 1.
- For quadratic/quadratic, the horizontal asymptote is the ratio of the x² coefficients. If the top is one degree higher, there is a slanted (oblique) asymptote found by division.
Sketching checklist
- Asymptotes.
- Crossings: x = 0 gives the y-intercept; top = 0 gives x-intercepts.
- Behaviour near each vertical asymptote (sign of y just left and just right).
- Join smoothly, never crossing a vertical asymptote.
Quadratic over quadratic: range without calculus
For y = (x² + …)/(x² + …), the curve may not take every y value. To find which values it takes:
- Put y = k and multiply out: k(bottom) = top.
- Rearrange to a quadratic in x.
- A real x exists only if the discriminant ≥ 0 (b² − 4ac ≥ 0).
- Solve that inequality in k. This gives the range, and the end values are the turning points' y-values.
Example: y = x/(x² + 1). k(x² + 1) = x → kx² − x + k = 0. Discriminant 1 − 4k² ≥ 0 → −1/2 ≤ k ≤ 1/2. So the range is −½ ≤ y ≤ ½, with turning points at y = ±½.
Conics: parabola, ellipse, hyperbola
Slicing a cone at different angles gives the conic sections.
- Parabola y² = 4ax: vertex (0, 0), focus (a, 0), directrix x = −a. Parametric form (at², 2at).
- Ellipse x²/a² + y²/b² = 1: crosses at (±a, 0) and (0, ±b). If a = b it is a circle. Parametric (a cos t, b sin t).
- Hyperbola x²/a² − y²/b² = 1: crosses at (±a, 0), never meets the y-axis, asymptotes y = ±(b/a)x. Parametric (a cosh t, b sinh t) or (a sec t, b tan t).
- Rectangular hyperbola xy = c²: asymptotes are the axes. Parametric (ct, c/t).
A focus is a special inside point. For the ellipse, foci are at (±c, 0) with c² = a² − b²; for the hyperbola c² = a² + b².
Graph transformations
- Replace x by (x − p): move right by p.
- Replace y by (y − q): move up by q.
- Replace x by x/s: stretch horizontally by factor s.
- Replace y by y/s: stretch vertically by factor s.
- Swap x and y: reflect in y = x.
- Replace x by −x (or y by −y): reflect in the y-axis (or x-axis).
Example: (x − 3)²/16 + (y + 1)²/4 = 1 is the ellipse x²/16 + y²/4 = 1 moved 3 right and 1 down. Its centre is (3, −1). Asymptotes, foci and vertices move with the curve.
Polynomial and rational inequalities
Never multiply both sides by something whose sign you don't know (like x − 2). Instead:
- Move everything to one side: f(x) > 0.
- Write as one fraction and factorise top and bottom.
- Find critical values: zeros of the top and of the bottom.
- Test the sign in each interval (or use a sketch).
- Choose the intervals you need. Bottom zeros are never included.
Another safe method: multiply both sides by (x − 2)², which is always positive.
Example: (x + 1)/(x − 2) > 2 → (x + 1 − 2x + 4)/(x − 2) > 0 → (5 − x)/(x − 2) > 0. Critical values 2 and 5. Positive between them: 2 < x < 5.
Try it: draw an ellipse with string
Push two pins into card about 8 cm apart. Tie a loop of string around them, pull it tight with a pencil and draw all the way round. You get an ellipse; the pins are the foci. Bring the pins closer: it becomes more like a circle. Then use the free play to stretch the ellipse with the slider.
Key formulas and definitions
- Vertical asymptote: bottom = 0; horizontal asymptote (equal degrees): ratio of leading coefficients
- Range of y = P(x)/Q(x): set y = k, discriminant ≥ 0
- Parabola y² = 4ax, focus (a, 0), directrix x = −a
- Ellipse x²/a² + y²/b² = 1, foci (±c, 0), c² = a² − b²
- Hyperbola x²/a² − y²/b² = 1, asymptotes y = ±(b/a)x, c² = a² + b²
- Rectangular hyperbola xy = c²
- Shift: x → x − p (right p), y → y − q (up q); stretch: x → x/s
Worked examples
1. State the asymptotes of y = (3x − 1)/(x + 4).
Bottom 0 at x = −4 → vertical asymptote x = −4. Leading coefficients 3/1 → horizontal asymptote y = 3.
2. Find where y = (2x − 6)/(x + 1) crosses the axes.
x = 0: y = −6/1 = −6, so (0, −6). Top = 0: x = 3, so (3, 0).
3. The parabola y² = 12x: find the focus and directrix.
4a = 12 → a = 3. Focus (3, 0), directrix x = −3.
4. Find the foci of x²/25 + y²/9 = 1.
a = 5, b = 3, c² = 25 − 9 = 16, c = 4. Foci (±4, 0).
5. Describe the transformation from x²/4 − y²/9 = 1 to (x + 2)²/4 − (y − 1)²/9 = 1, and give the new asymptotes.
Shift 2 left and 1 up. Old asymptotes y = ±(3/2)x become y − 1 = ±(3/2)(x + 2).
6. Find the range of y = (x² + 1)/(x² + x + 1) without calculus.
k(x² + x + 1) = x² + 1 → (k − 1)x² + kx + (k − 1) = 0. Discriminant k² − 4(k − 1)² ≥ 0 → (k − 2(k − 1))(k + 2(k − 1)) ≥ 0 → (2 − k)(3k − 2) ≥ 0 → 2/3 ≤ k ≤ 2. Range 2/3 ≤ y ≤ 2.
7. Solve x² − x − 6 < 0 and (x − 1)/(x + 3) ≤ 0.
(x − 3)(x + 2) < 0 → −2 < x < 3. For the fraction: critical values 1 and −3; negative between them, zero at x = 1 (included), x = −3 excluded → −3 < x ≤ 1.
Common mistakes
- Multiplying an inequality by (x − 2) without knowing its sign. Use critical values or multiply by (x − 2)².
- Including a value that makes the bottom zero in the answer of a rational inequality.
- Shifting the wrong way: (x − 3) moves the graph right, not left.
- Mixing ellipse and hyperbola foci: ellipse c² = a² − b², hyperbola c² = a² + b².