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Transformation of Graphs

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Vertical Shift $y = f(x) + k$

Shifts the graph of $y = f(x)$ $k$ units upward for $k > 0$ , and $k$ units downward for $k < 0$ .

Horizontal Shift $y = f(x + h)$

Shifts the graph of $y = f(x)$ $h$ units to the left for $h > 0$ , and $h$ units to the right for $h < 0$ .

Vertical Stretch $y = af(x)$ , $a > 1$

Stretches the graph of $y = f(x)$ vertically by a factor of $a$ away from the x-axis.

Vertical Compression $y = af(x)$ , $0 < a < 1$

Compresses the graph of $y = f(x)$ vertically by a factor of $a$ towards the x-axis.

Horizontal Stretch $y = f(bx)$ , $0 < b < 1$

Stretches the graph of $y = f(x)$ horizontally by a factor of $1/b$ away from the y-axis.

Horizontal Compression $y = f(bx)$ , $b > 1$

Compresses the graph of $y = f(x)$ horizontally by a factor of $1/b$ towards the y-axis.

Reflection Across Y-Axis $y = f(-x)$

Reflects the graph of $y = f(x)$ across the y-axis.

Reflection Across X-Axis $y = -f(x)$

Reflects the graph of $y = f(x)$ across the x-axis.

Rotation by 180 Degrees $y = -f(-x)$

Rotates the graph of $y = f(x)$ by 180 degrees.

Horizontal Translation $y = f(x-c)$

Translates the graph of $y = f(x)$ to the right by $c$ units.

Scaling Transformation $y = cf(ax + b) + d$

Scales and translates the graph of $y = f(x)$ by factors of $a, b, c, d$ .

Reflection Across the Line $y = x$ $y = f^{-1}(x)$

Reflects the graph of $y = f(x)$ across the line $y = x$ to obtain the graph of its inverse function.

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