Reciprocal Graphs (Edexcel IGCSE Maths)

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Reciprocal Graphs

What is a reciprocal graph?

• A reciprocal graph is of the form $y=\frac{a}{x}$ or $y=\frac{a}{x^{2}}$

  • These graphs do not have any y-intercepts
  • and do not have any roots
  • This means that the curves do not cross either the x- or y-axes

• The two basic reciprocal graphs have $a=1$

  • I.e. $y=\frac{1}{x}$ or $y=\frac{1}{x^{2}}$

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 if a is not equal to 1?

• You also need to recognise graphs of $y=\frac{a}{x}$ and $y=\frac{a}{x^{2}}$when $a\neq1$

  • In the graphs below the asymptotes are shown by dashed lines

• The sign of a shows where the curves are located

• The size of a shows how steep the curves are

  • The closer a is to 0 the more L-shaped the curves are

What if a constant is added to the equation?

• The reciprocal graphs, $y=\frac{a}{x}+b$ and $y=\frac{a}{x^{2}}+b$(where $a$ and $b$ are both constants)

  • are the same shapes as $y=\frac{a}{x}$ or $y=\frac{a}{x^{2}}$

  • but are shifted upwards by $b$ units

    • $y=\frac{1}{x}+2$ would be $y=\frac{1}{x}$ shifted up by 2 units
    • $y=\frac{4}{x^{2}}-3$ would be $y=\frac{4}{x^{2}}$ 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 at $y=6$
    • $y=\frac{1}{x^{2}}-7$ would have a horizontal asymptote at $y=-7$

  • The vertical asymptote remains along the y-axis

    • The equation of the vertical asymptote is $x=0$
    • $y=\frac{3}{x}+9$ and $y=\frac{6}{x^{2}}-5$ would both have vertical asymptotes at $x=0$

Worked Example Sketch the graph of $y=\frac{2}{x}-3$.

Answer:

The graph of $y=\frac{2}{x}-3$

• will have the same basic shape as $y=\frac{1}{x}$

  • (For a sketch, you don't need to worry abut the effect of the '2')

• but shifted down by 3 units because of the -3

  • (This means it will have an asymptote at $y=-3$)

It can be useful to sketch the asymptote first, to give you a 'guideline' for the rest of the sketch