transformations

• TODO watch + notes on all transformation videos 🔼 📅 2023-10-31

• there are 3 types of transformations:

  • Translations
  • dilations
  • reflections (often combined w/ dilations)

• Each of them can occur vertically or horizontally

Vertical:

Vertical Translations:

• translate means to shift
• a vertical translation of a point means only the y value changes
• e.g. image:
  • 
• translation of graphs:
  • each point $(x,f(x))$ maps to $(x,f(x)+c)$
  • e.g. to translate the graph $y=f(x)$ up by 2 would be $y=f(x)+2$

y=f(x)+c

#### Vertical Dilations: - a stretching or compressing in the y direction - by multiplying the y value by a constant - e.g. image: - ![[Pasted image 20231026141447.png|600]] - Dilation of graphs: - Each point $(x,f(x))$ maps to $(x,k \times f(x))$. - e.g. to stretch the graph $y=f(x)$ vertically by a factor of 2 would be $y=2f(x)$. - $$ y=k(f(x))

Horizontal:

Horizontal Translations:

• horizontal translation shifts the graph left or right.
• horizontal translation means only the x value changes
• e.g. image:
  • 
• translation of graphs:
  • e.g to horizontally translate a graph 3 left
  • a point $(x,f(x))$ maps to $(x-3,f(x))$
  • the function which has the input $x-3$ & the output $f(x)$:
    • $y=f(x+3)$
    • ^ which is the graph of $y=f(x)$ translated 3 to the left

y=f(x+b)

- ^ is $y=f(x)$ translated horizontally by $|b|$ - left if $b>0$ - right if $b<0$ #### Horizontal Dilations: - horizontal Dilation stretches or compresses in the x direction - by multiplying the x value by a constant (the y val doesn't change) - e.g. image: - ![[Pasted image 20231026143551.png|600]] - dilation of graphs: - e.g. to horizontally dilate a graph ($y=f(x)$) by a factor of 3 - a point $(x,f(x))$ maps to $(3x,f(x))$ - the function which has the input $3x$ & the output $f(x)$: - $y=f\left( \frac{1}{3}x \right)$ - ^ which is the graph of $y=f(x)$ dilated horizontally by a factor of 3 - $$ y=f(ax)

- ^ is $y=f(x)$ dilated horizontally by a factor of $\frac{1}{a}$
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