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      Transforming graphs: y = f(x + a)

      Lesson details

      Learning outcome

      I can recognise the effect of applying the transformation y = f(x + a) 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 horizontal translation in the opposite direction

      Keywords

      • Function - A function is a mathematical relationship that uniquely maps values of one set to the values of another set.

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

      Common misconception

      f$$(x)+5$$ is a translation of $$+5$$ in the $$y$$-direction so pupils may assume that f$$(x+5)$$ is a translation of $$+5$$ in the $$x$$-direction.

      Visually showing pupils, using graphing technology, that f$$(x+5)$$ is not a translation to the right will help. Showing them a comparison of the two tables of values for both functions shows them why this happens.

      Teacher tip

      Using technology as a visual aid to model this transformation helps pupil to see that the curve of any graph remains congruent and the only change is it's position horizontally. Having access to Desmos.com and the ability to project it for pupils will support learning.

      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.
      $$\text{f}(x)+a$$ is a of the graph of $$\text{f}(x)$$.

      Correct Answer: transformation, translation

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

      Correct Answer:$$\text{f}(x)+4$$,The output of $$\text{f}(x)$$ has $$4$$ added to it.

      The output of $$\text{f}(x)$$ has $$4$$ added to it.

      Correct Answer:$$\text{f}(x+4)$$,The input has $$4$$ added to it before the function is applied.

      The input has $$4$$ added to it before the function is applied.

      Correct Answer:$$4\text{f}(x)$$,The output of $$\text{f}(x)$$ is multiplied by $$4$$

      The output of $$\text{f}(x)$$ is multiplied by $$4$$

      Correct Answer:$$\text{f}(x)-4$$,The output of $$\text{f}(x)$$ has $$4$$ subtracted from it.

      The output of $$\text{f}(x)$$ has $$4$$ subtracted from it.

      Correct Answer:$$\text{f}(x-4)$$,The input has $$4$$ subtracted before the function is applied.

      The input has $$4$$ subtracted before the function is applied.

      Q3.
      Shape 'a' has been transformed to shape 'b' by translation of __________.

      An image in a quiz
      $$3$$ right, $$1$$ up.
      Correct answer: $$8$$ right, $$3$$ up.
      $$8$$ left, $$3$$ down.
      Correct answer: vector $$\begin{pmatrix}8 \\ 3\end{pmatrix}$$
      vector $$\begin{pmatrix}3 \\ 8\end{pmatrix}$$

      Q4.
      The transformation of the point $$(3,-2)$$ to the position $$(-4,-2)$$ could be described by __________.

      An image in a quiz
      translation by $$7$$ in the positive $$x$$-direction.
      Correct answer: translation by $$7$$ in the negative $$x$$-direction.
      translation by $$7$$ in the negative $$y$$-direction.
      Correct answer: translation by vector $$\begin{pmatrix}-7 \\ 0\end{pmatrix}$$
      reflection in the line $$x=-1$$

      Q5.
      The point $$(-1,13)$$ is translated $$+8$$ in the $$x$$-direction. What is its new position?

      Correct answer: $$(7,13)$$
      $$(-9,13)$$
      $$(-1,21)$$
      $$(-1,5)$$
      $$(8,13)$$

      Q6.
      The point $$(c,-d)$$ is translated by $$9$$ in the negative $$x$$-direction. What is its new position?

      $$(c,9-d)$$
      $$(c,-d-9)$$
      $$(-9,-d)$$
      Correct answer: $$(c-9,-d)$$
      $$(c+9,-d)$$

      6 Questions

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

      reflection
      rotation
      Correct answer: translation
      enlargement

      Q2.
      $$\text{f}(x+a)$$ is a transformation by translation of __________.

      $$a$$ in the $$x$$-direction.
      Correct answer: $$-a$$ in the $$x$$-direction.
      $$a$$ in the $$y$$-direction.
      $$-a$$ in the $$y$$-direction.

      Q3.
      $$y=\text{f}(x)$$ is graphed. Which of these images shows the transformation $$y=\text{f}(x-3)$$?

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

      Q4.
      $$y=\text{f}(x)$$ is graphed here. Which coordinate pairs will be on the transformation $$y=\text{f}(x+6)$$?

      An image in a quiz
      $$(-5,-4)$$
      Correct answer: $$(-11,-10)$$
      $$(4,-1)$$
      Correct answer: $$(-6,5)$$
      $$(0,-1)$$

      Q5.
      $$y=\text{f}(x)$$ is graphed. What transformation can we see graphed? $$y=$$

      An image in a quiz
      Correct Answer: f(x-7), y=f(x-7)

      Q6.
      $$y=\text{f}(x)$$ is graphed here. Which could the graph of the transformation be?

      An image in a quiz
      $$y=\text{f}(x+45)$$
      Correct answer: $$y=\text{f}(x+90)$$
      $$y=\text{f}(x-90)$$
      Correct answer: $$y=\text{f}(x-270)$$
      $$y=\text{f}(x+270)$$

      To help you plan your 11 maths lesson on: Transforming graphs: y = f(x + a), download all teaching resources for free and adapt to suit your pupils' needs...