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      Key features of a reciprocal graph

      Lesson details

      Learning outcome

      I can identify the key features of a reciprocal graph.

      Key learning points

      1. A reciprocal graph has a distinct shape.
      2. A reciprocal graph has two asymptotes.
      3. Reciprocal graphs of the form y = k/x have the axes as their asymptotes.

      Keywords

      • Asymptote - An asymptote is a line that a curve approaches but never touches.

      Common misconception

      All reciprocal graphs appear only in two quadrants and have both axes as asymptotes.

      Show pupils the graphs of the $$y={1\over{x}}$$, then $$y={1\over{x}}+1$$, then $$y={1\over{x+1}}$$ and get them to compare which forms have which axes as asymptotes.

      Teacher tip

      Making connections is really important when learning mathematics. Linking the fact that $$y={1\over{x}}+1$$ has an $$x$$-intercept because the equation $${1\over{x}}+1=0$$ has a solution empowers students. Challenge pupils to think 'Can the graph have a $$y$$-intercept?".

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      The of $$\frac{1}{9}$$ is 9.

      Correct Answer: reciprocal

      Q2.
      What is the value of $$y$$ when $$x=3$$ in the equation $$y=15x$$?

      Correct Answer: 45, y=45

      Q3.
      What is the value of $$y$$ when $$x=3$$ in the equation $$y={15\over{x}}$$?

      Correct Answer: 5, y=5

      Q4.
      What is the value of $$y$$ when $$x={1\over{2}}$$ in the equation $$y=12x$$?

      24
      12
      Correct answer: 6
      3

      Q5.
      What is the value of $$y$$ when $$x={1\over{2}}$$ in the equation $$y={12\over{x}}$$?

      6
      12
      Correct answer: 24
      48

      Q6.
      What is the value of $$y$$ when $$x=-{1\over{2}}$$ in the equation $$y={12\over{x}}$$? A calculator might be useful to check your calculation here.

      24
      6
      -6
      Correct answer: -24

      6 Questions

      Q1.
      Select the equations that can be plotted to give a reciprocal graph.

      $$y=2+x$$
      Correct answer: $$y=\frac{2}{x}$$
      $$y=x^2$$
      Correct answer: $$y=\frac{1}{2+x}$$
      $$y=2x$$

      Q2.
      __________ is a line that a curve approaches but never touches.

      Correct answer: An asymptote
      An axis
      An intersection
      An intercept
      A reciprocal

      Q3.
      Which of these statements is true for the graph of $$y=\frac{4}{x}+2$$

      The $$x$$-axis is an asymptote
      Correct answer: The $$y$$-axis is an asymptote
      Correct answer: The curve intersects the $$x$$-axis
      The curve intersects the $$y$$-axis

      Q4.
      Match each of curve to the equation of its horizontal asymptote

      Correct Answer:$$y=\frac{2}{x}$$,$$x$$-axis
      Correct Answer:$$y=\frac{3}{x}+2$$,$$y=2$$
      Correct Answer:$$y=\frac{2}{x}+3$$,$$y=3$$
      Correct Answer:$$y=\frac{3}{x}-2$$,$$y=-2$$
      Correct Answer:$$y=-\frac{2}{x}-3$$,$$y=-3$$

      Q5.
      The graph of $$y = \frac{1}{x-4}$$ has a vertical asymptote at $$x=$$

      Correct Answer: 4

      Q6.
      Which of these equations could be the equation of this curve?

      An image in a quiz
      $$y=\frac{3}{x}-2$$
      $$y=\frac{1}{x-2}+3$$
      $$y=\frac{2}{x}-3$$
      Correct answer: $$y=\frac{6}{x}-3$$
      $$y=\frac{6}{x}+3$$

      To help you plan your 10 maths lesson on: Key features of a reciprocal graph, download all teaching resources for free and adapt to suit your pupils' needs...