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Year 11
Higher

Transforming graphs: y = f(x + a)

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

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New
New
Year 11
Higher

Transforming graphs: y = f(x + a)

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

Download all resources
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Share activities with pupils
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Lesson details

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.

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.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), 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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Starter quiz

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

Exit quiz

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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: $$y=\text{f}(x-270)$$
$$y=\text{f}(x+270)$$