Translations of Functions
Translations involve shifting the graph of a function horizontally or vertically without changing its shape.
Vertical Translation
A vertical translation moves the graph of a function up or down on the coordinate plane. The general form is:
$y = f(x) + b$
where $b$ is the vertical shift.
- If $b > 0$, the graph shifts up by $|b|$ units.
- If $b< 0$, the graph shifts down by $|b|$ units.
Consider the function $f(x) = x^2$. The function $g(x) = x^2 + 3$ is a vertical translation of $f(x)$ by 3 units upward.
Horizontal Translation
A horizontal translation moves the graph of a function left or right on the coordinate plane. The general form is:
$y = f(x - a)$
where $a$ is the horizontal shift.
- If $a > 0$, the graph shifts right by $a$ units.
- If $a< 0$, the graph shifts left by $|a|$ units.
For the function $f(x) = x^2$, the function $h(x) = (x - 2)^2$ is a horizontal translation of $f(x)$ by 2 units to the right.
NoteIt's important to remember that for horizontal translations, the sign inside the parentheses is opposite to the direction of the shift.
Reflections of Functions
Reflections involve flipping the graph of a function over a particular axis.
Reflection in the x-axis
To reflect a function in the x-axis, we negate the function:
$y = -f(x)$
This flips the graph vertically, inverting all y-values.
Reflection in the y-axis
To reflect a function in the y-axis, we negate the input:
$y = f(-x)$
This flips the graph horizontally, reversing all x-values.
ExampleFor $f(x) = x^3$:
- $g(x) = -x^3$ is the reflection of $f(x)$ in the x-axis.
- $h(x) = (-x)^3$ is the reflection of $f(x)$ in the y-axis.
Vertical Stretch of a Function
A vertical stretch (or compression) changes the amplitude of a function. The general form is:
$y = pf(x)$
where $p$ is the scale factor.
- If $|p| > 1$, the graph is stretched vertically.
- If $0< |p| < 1$, the graph is compressed vertically.
