Single Transformations of Functions

Introduction

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

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

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

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

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: $f(x)={\sqrt {x}}$ (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 $f(x)=e^{x}$.

Even and Odd Functions

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

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

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

Compressions and Stretches

Given a function $f(x)$, a new function $g(x)=af(x)$, where $a$ is a constant, is a vertical stretch or vertical compression of the function $f(x)$.

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

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

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

Resources

• Intro to Transformations of Functions, Lumen Learning