Reciprocal Graphs (Cambridge International IGCSE Maths)

Reciprocal Graphs

What is a reciprocal graph?

• A reciprocal graph is of the form $y=\frac{a}{x}$

  • It does not have a y-intercept
  • and does not have any roots
  • This means that the curve does not cross either the x- or y-axes

What are the asymptotes on a reciprocal graph?

• An asymptote is a line on a graph that a curve becomes closer and closer to but never touches

  • These may be horizontal or vertical lines

• A reciprocal graph has two asymptotes

  • A horizontal asymptote along the x-axis (with equation $y=0$)

    • This is the limiting value of y when the value of x gets very large (either positive or negative)

  • A vertical asymptote along the y-axis (with equation $x=0$)

    • This shows the problem of trying to divide by zero

What about graphs of the form y = a/x + b ?

• The reciprocal graph, $y=\frac{a}{x}+b$ (where $a$ and $b$ are both constants)

  • is the same shape as $y=\frac{a}{x}$

  • but is shifted upwards by $b$ units

    • $y=\frac{1}{x}+2$ would be $y=\frac{1}{x}$ shifted up by 2 units
    • $y=\frac{4}{x}-3$ would be $y=\frac{4}{x}$ shifted down by 3 units

  • This means the horizontal asymptote also shifts up by $b$ units,

    • The equation of the horizontal asymptote is $y=b$
    • $y=\frac{5}{x}+6$ would have a horizontal asymptote of $y=6$
    • $y=\frac{1}{x}-7$ would have a horizontal asymptote of $y=-7$

  • The vertical asymptote remains along the y-axis

    • The equation of the vertical asymptote is $x=0$
    • $y=\frac{3}{x}+9$ would have a vertical asymptote of $x=0$

Worked Example The diagram below shows the graph of $y=\frac{2}{x}-4$.

(a) Write down the equation of the horizontal asymptote.

A reciprocal graph of the form $y=\frac{a}{x}+b$ will have a horizontal asymptote at $y=b$

$y=-4$

(b) Write down the equation of the vertical asymptote.

A reciprocal graph of the form $y=\frac{a}{x}+b$* will always have a vertical asymptote along the y-axis*

$x=0$