Translations involve shifting the graph of a function horizontally or vertically without changing its shape.
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.
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.
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.
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.
It's important to remember that for horizontal translations, the sign inside the parentheses is opposite to the direction of the shift.
Reflections involve flipping the graph of a function over a particular axis.
To reflect a function in the x-axis, we negate the function:
$y = -f(x)$
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