toPi 2e

8.01 + 8.02 - Translations

Properties1
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Translations

Translation is the process of moving a function.

Vertical Translations

𝑦=𝑓(𝑥)+𝑑𝑦→𝑦+𝑑

Take the function 𝑓(𝑥)=𝑥2.

When it is transformed to 𝑓(𝑥)=𝑥2+1 (𝑦=𝑓(𝑥)+1), the curve is shifted up by 1 unit.

Horizontal Translations

𝑦=𝑓(𝑥−𝑐)𝑥→𝑥+𝑐

Take the function 𝑓(𝑥)=𝑥3. When it is transformed to 𝑓(𝑥)=𝑥3−1 (𝑦=𝑓(𝑥)−1), the curve is shifted down by 1 unit.

Comparison

Tip

These pronumerals will be utilised for 8.06 + 8.07 - Combined Transformations. 𝑑 is for vertical translation, 𝑐 is for horizontal translation.

𝑦=𝑓(𝑥−𝑐)+𝑑
Vertical TranslationHorizontal Translation
𝑑:Vertical Translation𝑐:Horizontal Translation
𝑑>0=Shift Up

e.g.𝑑=3,𝑓(𝑥)+3		𝑐>0=Shift Right

e.g.𝑐=3,𝑓(𝑥−3)
𝑑<0=Shift Down

e.g.𝑑=−3,𝑓(𝑥)−3		𝑐<0=Shift Left

e.g.𝑐=3,𝑓(𝑥+3)

Example

Take the function 𝑓(𝑥)=𝑥2. Shift it up by three and shift it left by four.

Graph View

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