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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 of the function is transformed by a vertical translation

      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

      Pupils may not see the vertical translation depending on the function used to show the effect of $$a$$.

      Ensure pupils consider a range of functions so that they can see the movement of the graph.

      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 vertically. 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.
      When every point of a shape moves the same distance in the same direction the transformation is called a...

      Correct answer: translation
      reflection
      rotation
      enlargement

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

      Correct Answer:$$\text{f}(x)+11$$,The output of $$\text{f}(x)$$ has $$11$$ added to it.

      The output of $$\text{f}(x)$$ has $$11$$ added to it.

      Correct Answer:$$\text{f}(x+11)$$,The input has $$11$$ added to it before the function is applied.

      The input has $$11$$ added to it before the function is applied.

      Correct Answer:$$11\text{f}(x)$$,The output of $$\text{f}(x)$$ is multiplied by $$11$$

      The output of $$\text{f}(x)$$ is multiplied by $$11$$

      Correct Answer:$$\text{f}(x)-11$$,The output of $$\text{f}(x)$$ has $$11$$ subtracted from it.

      The output of $$\text{f}(x)$$ has $$11$$ subtracted from it.

      Correct Answer:$$\text{f}(x-11)$$,The input has $$11$$ subtracted before the function is applied.

      The input has $$11$$ subtracted before the function is applied.

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

      An image in a quiz
      $$2$$ right, $$1$$ down.
      Correct answer: $$4$$ right, $$6$$ down.
      $$4$$ left, $$6$$ up.
      vector $$\begin{pmatrix}4 \\ 6\end{pmatrix}$$
      Correct answer: vector $$\begin{pmatrix}4 \\ -6\end{pmatrix}$$

      Q4.
      For the function $$\text{f}(x)$$, $$\text{f}(8)=37$$, match these transformations of the function to their values.

      Correct Answer:$$\text{f}(8)+3$$,$$40$$

      $$40$$

      Correct Answer:$$3\text{f}(8)$$,$$111$$

      $$111$$

      Correct Answer:$$\text{f}(8)-3$$,$$34$$

      $$34$$

      Correct Answer:$$-3\text{f}(8)$$,$$-111$$

      $$-111$$

      Correct Answer:$$3\text{f}(8)-3$$,$$108$$

      $$108$$

      Q5.
      The point $$(4,1)$$ is translated 'up $$5$$'. What is it's new position?

      Correct answer: $$(4,6)$$
      $$(9,6)$$
      $$(4,5)$$
      $$(9,1)$$

      Q6.
      The point $$(-5,-a)$$ is translated 'down $$4$$'. What is it's new position?

      $$(-1,-a)$$
      $$(-9,-a)$$
      $$(-5,-4)$$
      Correct answer: $$(-5,-a-4)$$
      $$(-5,a-4)$$

      6 Questions

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

      reflection
      rotation
      Correct answer: translation
      enlargement

      Q2.
      $$\text{f}(x)+a$$ translates the graph by $$a$$ in the...

      Correct answer: $$y$$-direction
      positive $$y$$-direction
      $$x$$-direction
      positive $$x$$-direction

      Q3.
      $$y=\text{f}(x)$$ is graphed here. The second line represents what transformation of $$y=\text{f}(x)$$?

      An image in a quiz
      Correct answer: $$y=\text{f}(x)+3$$
      $$y=\text{f}(x)-3$$
      $$y=\text{f}(x)+6$$
      $$y=\text{f}(x)-6$$

      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
      $$(4,6)$$
      Correct answer: $$(4,-6)$$
      $$(-6,0)$$
      Correct answer: $$(0,-6)$$
      Correct answer: $$(5,-11)$$

      Q5.
      The points $$(-2, 24), (-1,-1), (0, 8)$$ lie on the graph of $$\text{f}(x)$$. Which points will lie on the graph of $$y=\text{f}(x)+12$$?

      $$(10,24)$$
      Correct answer: $$(-2,36)$$
      $$(12,8)$$
      Correct answer: $$(0,20)$$
      $$(-1,-13)$$

      Q6.
      $$(-c,d)$$ is on the graph of $$y=\text{f}(x)$$. What will the position of this point be if the graph is transformed by $$y=\text{f}(x)+a$$?

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

      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...