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Year 11
Higher

Transforming graphs: y = −f(x)

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

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New
New
Year 11
Higher

Transforming graphs: y = −f(x)

I can recognise the effect of applying the transformation y = −f(x) 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 reflection in the x axis

Keywords

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

Common misconception

Pupils can get confused between $$-$$f$$(x)$$ and f$$(-x)$$, which one is a reflection in the $$x$$-axis and which is a reflection in the $$y$$-axis?

Interpreting the notation is key. Remind pupils that $$-$$f$$(x)$$ is the output of f$$(x)$$ being multiplied by $$-1$$. Which coordinate is the output? The $$y$$ coordinates are altered from positive to negative, that is a reflection in which axis?

Visually showing pupils $$-$$f$$(x)$$ using graphing technology demonstrates that it is a reflection in the $$x$$-axis. Showing pupils the comparison of two tables of values explains why this occurs.
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.
$$\text{f}(x)+a$$ and $$\text{f}(x+a)$$ transform the graph of $$\text{f}(x)$$ by __________.
addition
Correct answer: translation
rotation
enlargement
Q2.
Match each piece of function notation to a description.
Correct Answer:$$\text{f}(x)+5$$,The output of $$\text{f}(x)$$ has $$5$$ added to it.

The output of $$\text{f}(x)$$ has $$5$$ added to it.

Correct Answer:$$\text{f}(x+5)$$,The input has $$5$$ added to it before the function is applied.

The input has $$5$$ added to it before the function is applied.

Correct Answer:$$5\text{f}(x)$$,The output of $$\text{f}(x)$$ is multiplied by $$5$$

The output of $$\text{f}(x)$$ is multiplied by $$5$$

Correct Answer:$$\text{f}(x)-5$$,The output of $$\text{f}(x)$$ has $$5$$ subtracted from it.

The output of $$\text{f}(x)$$ has $$5$$ subtracted from it.

Correct Answer:$$\text{f}(x-5)$$,The input has $$5$$ subtracted before the function is applied.

The input has $$5$$ subtracted before the function is applied.

Q3.
The transformation of the point $$(3,4)$$ to the position $$(-3,4)$$ could be described by __________.
An image in a quiz
Correct answer: translation by $$6$$ to the left
Correct answer: translation by vector $$\begin{pmatrix}-6 \\ 0\end{pmatrix}$$
translation by vector $$\begin{pmatrix}0 \\ -6\end{pmatrix}$$
reflection in the $$x$$-axis
Correct answer: reflection in the $$y$$-axis
Q4.
The transformation of the point $$(-2,5)$$ to the position $$(-2,-5)$$ could be described by __________.
An image in a quiz
Translation by vector $$\begin{pmatrix}-10 \\ 0\end{pmatrix}$$
Translation by vector $$\begin{pmatrix}0 \\ 10\end{pmatrix}$$
Correct answer: Translation by vector $$\begin{pmatrix}0 \\ -10\end{pmatrix}$$
Correct answer: reflection in the $$x$$-axis
reflection in the $$y$$-axis
Q5.
The point $$(7,-9)$$ is reflected in the $$x$$-axis. What is its new position?
$$(-7,-9)$$
$$(-7,9)$$
$$(-9,7)$$
Correct answer: $$(7,9)$$
Q6.
The point $$(-a,b)$$ is reflected in the $$x$$-axis. What is its new position?
$$(b,-a)$$
Correct answer: $$(-a,-b)$$
$$(a,-b)$$
$$(a,b)$$

Exit quiz

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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 __________.
in the $$y$$-axis
in the line $$y=x$$
Correct answer: in the $$x$$-axis
in the line $$y=-x$$
Q3.
$$y=\text{f}(x)$$ is graphed here. Which coordinate pairs will be on the transformed graph $$y=-\text{f}(x)$$?
An image in a quiz
$$(0,6)$$
$$(-6,0)$$
Correct answer: $$(1,-3)$$
Correct answer: $$(2,-2)$$
$$(-2,-2)$$
Q4.
$$y=\text{f}(x)$$ is graphed. Which of these images shows the transformation $$y=-\text{f}(x)$$?
An image in a quiz
Correct Answer: An image in a quiz
An image in a quiz
An image in a quiz
Q5.
Here is a table of values for an unknown function $$\text{f}(x)$$ Which coordinate pairs will be on the transformed graph $$y=-\text{f}(x)$$?
An image in a quiz
$$(3,-111)$$
$$(3,111)$$
Correct answer: $$(-2,40)$$
Correct answer: $$(0,-24)$$
$$(-1,-29)$$
Q6.
$$(-c,-d)$$ is on the graph of $$y=\text{f}(x)$$, $$c\neq{0}$$ and $$d\neq{0}$$. What will be the position of this point on the graph $$y=-\text{f}(x)$$?
$$(-c,-d)$$
$$(c,-d)$$
Correct answer: $$(-c,d)$$
$$(c,d)$$