Transforming graphs: combinations of transformations | Maths

Starter quiz

$y=\text{f}(x)+a$ , $y=\text{f}(x+a)$ , $y={a}\text{f}(x)$ , and $y=\text{f}(ax)$ are all ______ of $y=\text{f}(x)$ .

'transformations' ✓
Match the notation to its transformation.

$y=\text{f}(x)+a$

⇔

Translation by $a$ in the $y$ -direction. ✓

$y=\text{f}(x+a)$

⇔

Translation by $-a$ in the $x$ -direction. ✓

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

⇔

Reflection in the $x$ -axis. ✓

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

⇔

Reflection in the $y$ -axis. ✓

$y={a}\text{f}(x)$

⇔

Stretch by scale factor $a$ in the $y$ -direction. ✓

$y=\text{f}(ax)$

⇔

Stretch by scale factor $1\over{a}$ in the $x$ -direction. ✓
Match each piece of function notation to a description.

$2\text{f}(x)+3$

⇔

Output of $\text{f}(x)$ is multiplied by $2$ then $3$ is added. ✓

$3\text{f}(x)+2$

⇔

Output of $\text{f}(x)$ is multiplied by $3$ then $2$ is added. ✓

$3\text{f}(x+2)$

⇔

Input has $2$ added before output is multiplied by $3$ ✓

$2\text{f}(x+3)$

⇔

Input has $3$ added before output is multiplied by $2$ ✓

$2\text{f}(3x)$

⇔

Input is multiplied by $3$ before output is multiplied by $2$ ✓

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

⇔

Output of $\text{f}(x)$ is multiplied by $-2$ ✓
$y=\text{f}(x)$ is graphed. Which of these images shows the transformation $y=2\text{f}(x)$ ?
$y=\text{f}(x)$ is graphed. Which of these images shows the transformation $y=\text{f}(x-6)$ ?
$y=\text{f}(x)$ is graphed. Which of these images shows the transformation $y=-\text{f}(x)$ ?

Exit quiz

$-\text{f}(-x)$ will transform $\text{f}(x)$ by two __________.

translations

rotations

enlargements

reflections ✓
$y=\text{f}(x)$ is graphed. Which of these images shows the transformation $y=-2\text{f}(x)$ ?
$y=\text{f}(x)$ is graphed. Which of these images shows the transformation $y=2\text{f}(x)+2$ ?
$y=\text{f}(x)$ is graphed. Which of these images shows the transformation $y=-\text{f}(x+2)$ ?
$<Math>\text{f}(x)=\text{cos}(x)</Math>. Where will the marked point be after the transformation <Math>y=3\text{f}(2x)</Math>?$

$\text{f}(x)=\text{cos}(x)$ . Where will the marked point be after the transformation $y=3\text{f}(2x)$ ?

$(90,-3)$ ✓

$(-90,-3)$

$(-90,3)$

$(360,3)$

$(360,-3)$
$<Math>\text{f}(x)=\text{cos}(x)</Math>. Where will the marked point be after the transformation <Math>y=-\text{f}\left({1\over2}x\right)+1</Math>? Give your answer in the form <Math>(x,y)</Math>$

$\text{f}(x)=\text{cos}(x)$ . Where will the marked point be after the transformation $y=-\text{f}\left({1\over2}x\right)+1$ ? Give your answer in the form $(x,y)$

'(360, 2)' ✓

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Lesson Details

Key learning points

Desmos is an effective tool for showing the effects of this transformation
The actual function itself does not need to be known
The order of transformations is defined by the priority of operations

Common misconception

For the combined transformation $3$ f $(x-2)$ order does not matter so pupils may assume that order never matters.

For $3$ f $(x-2)$ the $3$ affects the output, the $y$ value, and the $-2$ affects the input, the $x$ value. Model to show that these can happen in either order. Then point out that $3$ f $(x)-2$ is two effects on the output.

Keywords

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