Graph Transformations

Graph transformations modify the appearance or position of a graph without altering its fundamental shape. Transformations can be categorised into translations, scaling (dilations), reflections, and combinations. These transformations are applied to the basic form of a function.

Translation

Translation shifts the graph horizontally or vertically without changing its shape.

• If $f(x)$ then $f(x-h)$ shifts the graph horizontally by $h$ units. If $h>0$, the graph is shifted right, and if $h<0$ then the graph is shifted left.
• If $f(x)$ then $f(x)+k$ shifts the graph vertically by $k$ units. If $k>0$, the graph is shifted upwards, and if $k<0$ then the graph is shifted downwards.
• For linear functions ($f(x)=mx+c$), the function is vertically translated by $c$.

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$y=3x+1$

$y=3x+2$

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Scaling

Scaling stretches or compresses the graph either horizontally or vertically.

• If $f(x)$ then $a \cdot f(x)$ stretches the graph vertically by a scale of $a$. If $a>1$, it is stretched vertically and if $0<a<1$, it compresses the graph vertically.
• If $f(x)$ then $f(bx)$ stretches the graph horizontally by a scale of $b$. If $b>1$, it is compressed horizontally and if $0<b<1$, it stretches the graph horizontally.

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*$y=x^2, y=2x^2, y=4x^2$ , red, blue green respectively *

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Reflection

Reflection flips the graph across an axis.

• If $y=-f(x)$ inverts all $y$ values, which flips the graph upside down.
• If $y=f(-x)$ inverts all $x$ values.

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$y=x^2, y=-x^2$ red, purple respectively

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