Shifting functions along the x axis

A horizontal shift slides the graph sideways while keeping its exact shape.
Replacing $x$ with $x + k$ moves the graph left by $k$; replacing $x$ with $x - k$ moves it right by $k$.
The sign works opposite to what you might expect.

The figure shows a graph and its horizontal shifts.
Replacing $x$ with $x + 2$ slides the curve $2$ units left, while $x - 2$ slides it $2$ units right.
The plus sign shifts left and the minus sign shifts right.

Every point moves by the same amount, so the vertex and the $x$-intercepts shift too.
If a curve crosses the $x$-axis at $-3$ and $1$, then $f(x + 2)$ crosses at $-5$ and $-1$.
Each intercept moves $2$ units to the left.

This works for any function, not just parabolas.
Shifting $y = 3^x$ left by $4$ units gives $y = 3^{x + 4}$.
You always replace $x$ with $x + k$ for a left shift.

The graph of $y = 5^x$ is shifted $2$ units to the left. What is the new equation?
Shifting left by $2$ means replacing $x$ with $x + 2$.
So the new equation is $y = 5^{x + 2}$.

A curve crosses the $x$-axis at $-2$ and $4$. Where does $f(x + 3)$ cross the $x$-axis?
$f(x + 3)$ shifts the graph $3$ units left, so each intercept moves $3$ left.
So the new $x$-intercepts are $-5$ and $1$.

$f(x)$ has its lowest point at $x = -0.5$. Where is the lowest point of $f(x + 2)$?
The lowest point of $f$ happens when the input to $f$ equals $-0.5$.
For $f(x + 2)$, the input is $x + 2$, so set $x + 2 = -0.5$.
Solving gives $x = -0.5 - 2 = -2.5$.