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      Transforming graphs: y = f(−x)

      Lesson details

      Learning outcome

      I can recognise the effect of applying the transformation y = f(−x) to a graph.

      Key learning points

      1. Desmos is an effective tool for showing the effects of this transformation
      2. The actual function itself does not need to be known
      3. The graph is transformed by a reflection in the y axis

      Keywords

      • Transformation - A transformation is a process that may change the size, orientation or position of a shape.

      Common misconception

      Pupils mix-up $$-$$f$$(x)$$ and f$$(-x)$$. Which is a reflection in the $$x$$-axis? Which is in the $$y$$-axis?

      f$$(-x)$$ is the input being multiplied by $$-1$$ before the function is applied. Which coordinate is the input? The $$x$$ coordinates change from positive to negative so this must be a reflection in the $$y$$ axis.

      Teacher tip

      Visually showing pupils f$$(-x)$$ using graphing technology demonstrates that it is a reflection in the $$y$$ axis. Showing pupils the comparison of two tables of values explains why this occurs.

      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.
      $$-\text{f}(x)$$ transforms the graph of $$\text{f}(x)$$ by __________ $$x$$-axis.

      Correct answer: reflection in the
      rotation around the
      translation from the
      enlargement from the

      Q2.
      Match each piece of function notation to a description.

      Correct Answer:$$\text{f}(x)+1$$,The output of $$\text{f}(x)$$ has $$1$$ added to it.

      The output of $$\text{f}(x)$$ has $$1$$ added to it.

      Correct Answer:$$\text{f}(x+1)$$,The input has $$1$$ added to it before the function is applied.

      The input has $$1$$ added to it before the function is applied.

      Correct Answer:$$-\text{f}(x)$$,The output of $$\text{f}(x)$$ is multiplied by $$-1$$

      The output of $$\text{f}(x)$$ is multiplied by $$-1$$

      Correct Answer:$$\text{f}(x)-1$$,The output of $$\text{f}(x)$$ has $$1$$ subtracted from it.

      The output of $$\text{f}(x)$$ has $$1$$ subtracted from it.

      Correct Answer:$$\text{f}(x-1)$$,The input has $$1$$ subtracted before the function is applied.

      The input has $$1$$ subtracted before the function is applied.

      Q3.
      The transformation of shape 'a' to shape 'b' could be described by __________.

      An image in a quiz
      rotation $$90$$ degrees clockwise around the origin
      rotation $$90$$ degrees anti-clockwise around the origin
      Correct answer: reflection in the $$x$$-axis
      reflection in the $$y$$-axis
      Correct answer: translation by vector $$\begin{pmatrix}0 \\ -10\end{pmatrix}$$

      Q4.
      The transformation of the point $$(-3,2)$$ to the position $$(-3,-2)$$ could be described by __________.

      An image in a quiz
      translation by vector $$\begin{pmatrix}-4 \\ 0\end{pmatrix}$$
      Correct answer: translation by vector $$\begin{pmatrix}0 \\ -4\end{pmatrix}$$
      Correct answer: reflection in the $$x$$-axis
      reflection in the $$y$$-axis
      rotation $$90$$ degrees anti-clockwise around the origin

      Q5.
      The point $$(7,-9)$$ is reflected in the $$y$$-axis. What is its new position?

      $$(-9,7)$$
      $$(9,-7)$$
      Correct answer: $$(-7,-9)$$
      $$(7,9)$$
      $$(-7,9)$$

      Q6.
      The point $$(-c,d)$$ is reflected in the $$y$$-axis. What is its new position?

      $$(d,-c)$$
      $$(c,-d)$$
      $$(-c,-d)$$
      Correct answer: $$(c,d)$$

      6 Questions

      Q1.
      $$\text{f}(-x)$$ is a transformation by __________.

      translation
      rotation
      Correct answer: reflection
      enlargement

      Q2.
      $$\text{f}(-x)$$ is a transformation by reflection __________.

      Correct answer: in the $$y$$-axis
      in the line $$y=x$$
      in the $$x$$-axis
      in the line $$y=-x$$

      Q3.
      $$y=\text{f}(x)$$ is graphed. Which of these images shows the graph of $$y=\text{f}(-x)$$?

      Correct Answer: An image in a quiz
      An image in a quiz
      An image in a quiz
      An image in a quiz

      Q4.
      Which point will remain invariant by the transformation $$y=\text{f}(-x)$$? Give your answer in the form $$(x,y)$$

      An image in a quiz
      Correct Answer: (0, 1), (0,1)

      Q5.
      Here is a table of values for an unknown function $$\text{f}(x)$$. Which coordinate pairs will be on the graph $$y=\text{f}(-x)$$?

      An image in a quiz
      $$(-3,73)$$
      Correct answer: $$(2,-24)$$
      $$(1,-1)$$
      Correct answer: $$(0,8)$$
      Correct answer: $$(-1,3)$$

      Q6.
      $$(e,f)$$ is on the graph of $$y=\text{f}(x)$$, $$e\neq{0}$$ and $$f\neq{0}$$. What will be the position of this point on the graph $$y=\text{f}(-x)$$?

      Correct answer: $$(-e,f)$$
      $$(e,-f)$$
      $$(e,f)$$
      $$(-e,-f)$$

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