Reciprocal Graph y = k/x

Statement

The reciprocal function is given by:

\[ y = \frac{k}{x}, \quad k \neq 0. \]

Its graph is a rectangular hyperbola with two branches, one in the first and third quadrants if \(k > 0\), and one in the second and fourth quadrants if \(k < 0\). The asymptotes are the coordinate axes \(x = 0\) and \(y = 0\).

Why it’s true

• Division by zero is undefined, so the function cannot cross \(x = 0\). This makes \(x = 0\) a vertical asymptote.
• As \(x \to \infty\), \(y \to 0\). Similarly, as \(x \to -\infty\), \(y \to 0\). Therefore, \(y = 0\) is a horizontal asymptote.
• The symmetry of the graph arises because multiplying numerator and denominator by \(-1\) shows the function is odd: \(f(-x) = -f(x)\).

Recipe (how to use it)

1. Identify the constant \(k\).
2. Plot a few values of \(y = k/x\) for both positive and negative \(x\).
3. Sketch the hyperbola in the correct quadrants.
4. Mark the asymptotes \(x=0\) and \(y=0\).

Spotting it

Whenever the equation has the form \(y = k/x\) or \(y = \text{constant}/x\), it’s a reciprocal graph. It never crosses the axes, only approaches them.

Common pairings

• Transformations such as \(y = \frac{k}{x} + c\) (moves the graph up or down).
• Sketching functions in exam questions.
• Inverse proportionality in real-life contexts (e.g., time vs speed).

Mini examples

1. \(y = \tfrac{4}{x}\) → passes through (1,4), (2,2), (4,1), with asymptotes x=0, y=0.
2. \(y = -\tfrac{3}{x}\) → branches in 2nd and 4th quadrants.

Pitfalls

• Forgetting asymptotes — the graph never touches the axes.
• Mixing quadrants — positive \(k\) gives quadrants I and III; negative \(k\) gives quadrants II and IV.
• Only plotting positive \(x\)-values — must include negatives as well.

Exam strategy

• Always sketch at least 3–4 points in each branch.
• Clearly mark asymptotes on your diagram.
• Check the sign of \(k\) to choose correct quadrants.

Summary

The reciprocal function is a standard non-linear graph that shows inverse proportionality. It has asymptotes at both axes, with branch placement determined by the sign of \(k\). It is a frequent exam topic in graph sketching and real-world modelling.