Problem solving with graph transformations

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Lesson details

Learning outcome

I can use my knowledge of graph transformations to solve problems.

Key learning points

Your knowledge of identifying transformations of shapes can be applied to graphs
This allows you to identify what transformations have taken place
Once the transformations have been identified, you can write this in function notation

Keywords

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 link between transformations and manipulating algebraic expressions.

Be explicit in highlighting that the graph of $$y=x^2+10x+22$$ is the graph of $$y=(x+5)^2-3$$ which is the graph of $$y=x^2$$ transformed by the translation, "$$5$$ units left, $$3$$ units down".

Teacher tip

Ask pupils questions like, "What do we know about the quadratic $$y=x^2+6x-1$$?" They should now be able to add, "It is a transformation of $$y=x^2$$ by translation, "$$3$$ units left, $$10$$ units down".

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).

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Worksheet

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Prior knowledge starter quiz

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6 Questions

Q1.
Match the transformed functions of $$ \text {f}(x)$$ to their descriptions.

Correct Answer:$$ \text {f}(x) + 5$$,Translation of 5 in the positive $$y$$ direction

Translation of 5 in the positive $$y$$ direction

Correct Answer:$$ \text {f}(x+5) $$,Translation of 5 in the negative $$x$$ direction

Translation of 5 in the negative $$x$$ direction

Correct Answer:$$ \text {f}(5x)$$,Stretch of $$1 \over 5$$ in the $$x$$ direction

Stretch of $$1 \over 5$$ in the $$x$$ direction

Correct Answer:$$5 \text {f}(x) $$,Stretch of 5 in the $$y$$ direction

Stretch of 5 in the $$y$$ direction

Correct Answer:$$ -\text {f}(x) $$,Reflection in the $$x$$ axis

Reflection in the $$x$$ axis

Correct Answer:$$ \text {f}(-x) $$,Reflection in the $$y$$ axis

Reflection in the $$y$$ axis

Q2.
The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?

$$ \text {f}(x + 2)$$

$$ \text {f}(x - 2)$$

Correct answer: $$ \text {f}(2x)$$

$$ 2\text {f}(x)$$

$$ \text {f}\left({1\over2}x\right)$$

Q3.
The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?

Correct answer: $$ \text {f}(x + 2)$$

$$ \text {f}(x - 2)$$

$$ \text {f}(x) - 2$$

$$ -\text {f}(x)$$

$$ \text {f}(-x)$$

Q4.
The graph of $$y= \text{g} (x)$$ is labelled. What other function of $$x$$ has been graphed?

$$ \text {g}(x) - 8$$

$$ \text {g}(x- 8)$$

Correct answer: $$ -\text {g}(x)$$

$$ \text {g}(-x)$$

Q5.
An __________ is a line which a curve approaches but never touches.

Correct answer: asymptote

axis

exponential

image

invariant line

Q6.
Which of these is equivalent to $$x^2 - 4x + 6$$?

$$(x-2)^2-4$$

Correct answer: $$(x-2)^2+2$$

$$(x-2)^2+10$$

$$(x-4)^2-10$$

$$(x-4)^2+6$$

Assessment exit quiz

Download exit quiz questions (PDF)

6 Questions

Q1.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

$$y=x^2 +3$$

$$y=x^2 -3$$

$$y=(x+3)^2$$

Correct answer: $$y=(x-3)^2$$

Q2.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Correct answer: $$y=x^2 + 4x$$

$$y=x^2 - 4x$$

$$y=x^2 - 8x$$

$$y=x^2 + 4x + 8$$

$$y=x^2 + 4x - 8$$

Q3.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Correct answer: $$y=-4x^2$$

$$y=-2x^2$$

$$y=2x^2$$

$$y=4x^2$$

Q4.
The graph of $$y=x^2-2x$$ has been transformed to get a new graph. What is the equation of this new graph?

$$y=2x^2-2$$

$$y=2x^2+2$$

$$y=2x^2-4x+2$$

Correct answer: $$y=2x^2-8x+6$$

$$y=4x^2-8x+3$$

Q5.
The graph of $$\text{f}(x)=2^x + 1$$ has an asymptote at $$y=1$$. Match up the transformations with the equation of the asymptote after the transformation.

Correct Answer:$$\text {f}(x+3)$$,$$y=1$$

$$y=1$$

Correct Answer:$$\text {f}(x)+3$$,$$y=4$$

$$y=4$$

Correct Answer:$$3\text {f}(x)$$,$$y=3$$

$$y=3$$

Correct Answer:$$-\text {f}(x)$$,$$y=-1$$

$$y=-1$$

Q6.
The graph of $$\text{f}(x)= \text{tan}(x)$$ where $$-90 \le x \le 90$$ has asymptotes at $$x=-90$$ and $$x=90$$, which of these transformations will not change the position of the asymptotes ?

Correct answer: $$-\text{f}(x)$$

Correct answer: $$3\text{f}(x)$$

$$\text{f}(x+30)$$

Correct answer: $$\text{f}(x) + 3$$

$$\text{f}(3x)$$

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Problem solving with graph transformations

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Match the transformed functions of $$ \text {f}(x)$$ to their descriptions.

Q2.The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?

Q3.The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?

Q4.The graph of $$y= \text{g} (x)$$ is labelled. What other function of $$x$$ has been graphed?

Q5.An __________ is a line which a curve approaches but never touches.

Q6.Which of these is equivalent to $$x^2 - 4x + 6$$?

Assessment exit quiz

6 Questions

Q1.The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Q2.The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Q3.The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Q4.The graph of $$y=x^2-2x$$ has been transformed to get a new graph. What is the equation of this new graph?

Q5.The graph of $$\text{f}(x)=2^x + 1$$ has an asymptote at $$y=1$$. Match up the transformations with the equation of the asymptote after the transformation.

Q6.The graph of $$\text{f}(x)= \text{tan}(x)$$ where $$-90 \le x \le 90$$ has asymptotes at $$x=-90$$ and $$x=90$$, which of these transformations will not change the position of the asymptotes ?

How to plan a lesson using our resources

Q1.
Match the transformed functions of $$ \text {f}(x)$$ to their descriptions.

Q2.
The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?

Q3.
The graph of $$y= \text{f}(x)$$ is labelled. What other function of $$x$$ has been graphed?

Q4.
The graph of $$y= \text{g} (x)$$ is labelled. What other function of $$x$$ has been graphed?

Q5.
An __________ is a line which a curve approaches but never touches.

Q6.
Which of these is equivalent to $$x^2 - 4x + 6$$?

Q1.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Q2.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Q3.
The graph of $$y=x^2$$ has been transformed to get a new graph. What is the equation of this new graph?

Q4.
The graph of $$y=x^2-2x$$ has been transformed to get a new graph. What is the equation of this new graph?

Q5.
The graph of $$\text{f}(x)=2^x + 1$$ has an asymptote at $$y=1$$. Match up the transformations with the equation of the asymptote after the transformation.

Q6.
The graph of $$\text{f}(x)= \text{tan}(x)$$ where $$-90 \le x \le 90$$ has asymptotes at $$x=-90$$ and $$x=90$$, which of these transformations will not change the position of the asymptotes ?