Single Transformations of Functions

Introduction

• Vertical shift.png

  Vertical shift: $f(x)=x^{2}$ (red) and $g(x)=x^{2}+4$ (blue)

• Horizontal shift.png

  Horizontal shift: $f(x)=sin(x)$ (red) and $g(x)=sin(x+{\frac {\pi }{2}})$ (blue)

• Vertical reflection.png

  Vertical reflection: $f(x)={\sqrt {x}}$ (red) and $g(x)=-{\sqrt {x}}$ (blue)

• Horizontal reflection.png

  Horizontal reflection: $f(x)=e^{x}$ (red) and $g(x)=e^{-x}$ (blue)

• Vertical stretch compression.png

  Vertical stretch/compression: $f(x)=x^{2}$ (red), $g(x)={\frac {1}{2}}x^{2}$ (blue, vert. compression), $h(x)=2x^{2}$ (green, vert. stretch), and $j(x)=-2x^{2}$ (black, vert. stretch and vert. reflection)

• Horizontal stretch compression.png

  Horizontal stretch/compression: $f(x)={\sqrt {x}}$ (red), $g(x)={\sqrt {2x}}$ (blue, horiz. compression), $h(x)={\sqrt {{\frac {1}{2}}x}}$ (green, horiz. stretch), and $j(x)={\sqrt {-2x}}$ (black, horiz. compression and horiz. reflection)

Translations

One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function $g(x)=f(x)+k$, the function $f(x)$ is shifted vertically k units. For example, $g(x)=x^{2}+4$ is the function $f(x)=x^{2}$ shifted up by 4 units. $g(x)=sin(x)-7.7$ is the function $f(x)=sin(x)$ shifted down by 7.7 units.

Given a function f, a new function $g(x)=f(x-h)$, where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, $g(x)=(x-3)^{2}$ is the graph of $f(x)=x^{2}$ shifted 3 units to the right. $g(x)=sin(x+{\frac {\pi }{2}})$ is the function $f(x)=sin(x)$ shifted ${\frac {\pi }{2}}$ units to the left.

Reflections

Given a function $f(x)$, a new function $g(x)=-f(x)$ is a vertical reflection of the function $f(x)$, sometimes called a reflection about (or over, or through) the x-axis. For example, $g(x)=-{\sqrt {x}}$ is a vertical reflection of the function $f(x)={\sqrt {x}}$.

Given a function $f(x)$, a new function $g(x)=f(-x)$ is a horizontal reflection of the function $f(x)$, sometimes called a reflection about the y-axis. For example, $g(x)=e^{-x}$ is a horizontal reflection of 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) = e^x } .

Even and Odd Functions

A function f is even if for all values of x, 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) = f(-x) } ; that is, a 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) } is even if its horizontal reflection 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 identical to itself. For example, 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) = x^2 } is an even function since 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) = (-x)^2 = (-1)^2(x)^2 = x^2 = f(x)} .

A function f is odd if for all values of x, 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) = -f(-x) } ; that is, a 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) } is odd if a horizontal reflection and vertical reflection of the function results in the same function. For example, 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) = x^3 } is an odd function since 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) = -(-x)^3 = (-1)(-1)^3(x)^3 = x^3 = f(x)} .

If a function satisfies neither of these conditions, then it is neither even nor odd. For example, 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) = x^2 + x } is neither even nor odd because 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) = (-x)^2 + (-x) = x^2 - x } , which is not equal to 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) } , and 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) = -(-x)^2 - (-x) = -x^2 + x } , which is also not equal to $f(x)$.

Compressions and Stretches

Given a 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) } , a new 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 g(x) = af(x) } , where 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 a } is a constant, is a vertical stretch or vertical compression of 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) } .

• 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 a > 1 } , then the graph will be stretched.
• 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 0 < a < 1 } , then the graph will be compressed.
• If a < 0, then there will be a vertical stretch or compression of a factor of 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 |a| } , along with a vertical reflection.

Given a 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) } , a new 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 g(x) = f(bx) } , where 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 b } is a constant, is a horizontal stretch or horizontal compression of 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) } .

• 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 b > 1 } , then the graph will be horizontally compressed by a factor of 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 \frac{1}{b} } .
• 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 0 < b < 1 } , then the graph will be horizontally stretched by a factor of 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 \frac{1}{b} } .
• If b < 0, then there will be a horizontal stretch or compression of a factor of 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 \Big|\frac{1}{b} \Big| } , along with a vertical reflection.

Resources

• Intro to Transformations of Functions, Lumen Learning

Licensing

Content obtained and/or adapted from:

• Transformations of Functions, Lumen Learning College Algebra under a CC BY-SA license