Practice Maths

Hyperbolas

Key Ideas

Key Terms

hyperbola
Has equation y = k/x (equivalently xy = k). It has two separate branches.
Asymptotes
Lines the graph approaches but never touches. For y = k/x: asymptotes are x = 0 (y-axis) and y = 0 (x-axis).
k > 0
Branches are in quadrants 1 and 3. If k < 0: branches are in quadrants 2 and 4.
Domain
X ≠ 0 (all real x except 0). Range: y ≠ 0 (all real y except 0).
Feature y = k/x  (k > 0) y = k/x  (k < 0)
Branches inQuadrants 1 and 3Quadrants 2 and 4
Asymptotesx = 0 and y = 0
Domainx ≠ 0 (all real except 0)
Rangey ≠ 0 (all real except 0)
x- and y-interceptsNone (curve never touches either axis)

Graph of y = 2/x with key features marked:

x = 0 y = 0 x y −4 −3 −2 −1 1 2 3 4 4 3 2 1 −1 −2 −3 −4 (1, 2) (2, 1) y = 2/x
— — asymptotes (x = 0, y = 0) key points

Worked Example — Evaluate and find k

For y = 6/x, find y when x = 3. Then find x when y = −2.

y(3) = 6/3 = 2.

−2 = 6/x ⇒ x = 6/(−2) = −3.

Finding k: If the hyperbola passes through (4, −3), then y = k/x ⇒ −3 = k/4 ⇒ k = −12. Equation: y = −12/x.

Hot Tip — Shifted Hyperbola: y = k/(x − h) + c has the same shape but shifted. Asymptotes become x = h and y = c. Example: y = 3/(x − 2) + 1 has asymptotes x = 2 and y = 1, and branches in the direction determined by k = 3 > 0 relative to the new origin (2, 1).

Why Hyperbolas Arise

A hyperbola y = k/x describes any situation where two quantities multiply to give a constant. For example: if you travel a fixed distance and speed × time = constant, then time = distance/speed, which is a hyperbola in time vs speed. This is called inverse proportion — as one quantity increases, the other decreases proportionally.

The constant k is called the constant of proportionality. The equation xy = k (rearranged) makes this clear: at every point on the curve, the product of the coordinates equals k.

The Two Branches

When k > 0: both x and y have the same sign (both positive in Q1, both negative in Q3). The branches curve through quadrants 1 and 3.

When k < 0: x and y have opposite signs (Q2: x negative, y positive; Q4: x positive, y negative). The branches curve through quadrants 2 and 4.

The larger |k|, the further the curve sits from the origin — the branches are more stretched outward. Smaller |k| means the curve is tucked closer to the axes.

Asymptotes Explained

As x approaches 0 (from either side), y = k/x becomes very large in magnitude — the curve shoots up (or down) toward infinity, getting closer and closer to the y-axis but never touching it. This is the vertical asymptote x = 0.

As x becomes very large (positive or negative), k/x approaches 0 — the curve gets closer and closer to the x-axis but never reaches it. This is the horizontal asymptote y = 0.

Shifted Hyperbolas: y = k/(x − h) + c

Shifting the hyperbola moves both asymptotes. The vertical asymptote shifts from x = 0 to x = h (where the denominator equals zero). The horizontal asymptote shifts from y = 0 to y = c.

The branches now radiate from the new “centre” (h, c) rather than the origin. To sketch: draw the new asymptotes first as dashed lines, then draw the two branches curving toward them.

Example: y = −4/(x + 1) − 2. Here h = −1, c = −2, k = −4. Asymptotes: x = −1, y = −2. Since k < 0: branches in Q2/Q4 relative to the centre (−1, −2).

Common Mistakes:
  • Thinking y = k/x has x-intercepts or y-intercepts — it never crosses either axis.
  • Confusing k > 0 (Q1/Q3 branches) and k < 0 (Q2/Q4 branches).
  • For y = k/(x−h)+c, finding asymptotes as x=h incorrectly (check: the denominator is zero when x=h, so x=h is the vertical asymptote).

Mastery Practice

  1. For each hyperbola, state the asymptotes, the quadrants containing the branches, the domain, and the range. Fluency

      Equation Asymptotes Branches in Domain Range
    (a)y = 5/x    
    (b)y = −3/x    
    (c)xy = 8    
    (d)xy = −6    

  2. For the hyperbola y = 12/x: Fluency

    1. Find y when x = 3, x = −4, and x = 1.5.
    2. Find x when y = 6, y = −2, and y = 24.
    3. Does the point (3, 3) lie on this hyperbola? Justify.
    4. Find a point with x-coordinate 5 that lies on y = 12/x.

  3. Each hyperbola has the form y = k/x. Find k and write the equation. Fluency

    1. The curve passes through (3, 4).
    2. The curve passes through (−2, 7).
    3. The curve passes through (5, −3).
    4. The curve passes through (−6, −2).

  4. For each shifted hyperbola y = k/(x − h) + c, state the asymptotes. Fluency

    1. y = 2/(x − 3) + 1
    2. y = −5/(x + 2) − 4
    3. y = 3/x + 7
    4. y = 4/(x − 1)

  5. Comparing hyperbolas. Understanding

    1. Describe how y = 10/x differs from y = 1/x in appearance.
    2. Describe how y = −4/x differs from y = 4/x in appearance.
    3. For the hyperbola y = k/x, the point (2, 6) lies on the curve. Without finding k, explain why the point (6, 2) must also lie on the same curve.
    4. State the value of k for which y = k/x passes through the point (−3, −3).

  6. Speed and travel time. Understanding

    Travel. A car travels a fixed distance of 240 km. The travel time T (hours) depends on the average speed v (km/h), modelled by T = 240/v.
    1. Find the travel time at speeds of 60, 80, and 120 km/h.
    2. What speed is needed to complete the trip in 3 hours?
    3. Describe in words what happens to the travel time as speed doubles.
    4. Is there a speed at which the travel time equals zero? Explain using the equation.

  7. Read from the graph. Understanding

    The graph of a hyperbola y = k/x is shown below. One point is labelled.

    x y −3 −2 1 2 3 2 1 −1 −2 (2, 3)
    1. In which quadrant(s) do the branches lie? What does this tell you about the sign of k?
    2. State the equations of the asymptotes (shown as dashed lines).
    3. Use the labelled point (2, 3) to find the value of k.
    4. Write the equation of the hyperbola.

  8. Asymptotes and behaviour near them. Understanding

    Analysis. The hyperbola y = 6/(x − 2) + 3.
    1. State the asymptotes.
    2. Find y when x = 5, x = 0, and x = −1.
    3. Describe what happens to y as x approaches 2 from the right (i.e. x = 2.1, 2.01, 2.001, …).
    4. Describe what happens to y as x becomes very large and positive.

  9. Finding the equation. Problem Solving

    Algebraic problem. A hyperbola of the form y = k/(x − h) + c has vertical asymptote x = 4, horizontal asymptote y = −1, and passes through the point (6, 4).
    1. Use the asymptote information to write the equation in the form y = k/(x − 4) − 1.
    2. Substitute the point (6, 4) to find k.
    3. Write the complete equation and find the value of y when x = 5.

  10. Gas pressure and volume. Problem Solving

    Science (Boyle’s Law). At constant temperature, the pressure P (kPa) of a gas and its volume V (litres) are related by PV = k. At a pressure of 200 kPa, the volume is 3 litres.
    1. Find k and write the equation for P in terms of V.
    2. Find the pressure when the volume is 2 litres, 4 litres, and 6 litres.
    3. What volume corresponds to a pressure of 150 kPa?
    4. If the volume is halved, what happens to the pressure? Justify using the equation.
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