Multiple Transformations of Functions

See Single Transformations of Functions for more information on translating, reflecting, compressing, and stretching functions.

Combining Functions

Combining vertical and horizontal shifts

Vertical and horizontal shift: f(x) = x^2 + x (red) and g(x) = (x - 3)^2 + (x - 3) + 5 (blue; f(x) shifted 3 units right and 5 units up)

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } is some function, then $g(x)=f(x-h)+k$ is the function $f(x)$ shifted h units horizontally (to the right for h > 0 and to the left for h < 0) and k units vertically (up for k > 0 and down for k < 0). For example, $g(x)=(x-3)^{2}+(x-3)+5$ is the function $f(x)=x^{2}+x$ shifted 3 units to the right and 5 units up. $g(x)={\sqrt {x+2}}-6$ is the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \sqrt{x} } shifted 2 units to the left and 6 units down.