Shifting functions along the y-axis

Adding a constant outside the function, as in $f(x) + c$, shifts the entire graph up or down.
A positive $c$ raises every point by $c$; a negative $c$ lowers it.

Because every point moves by the same amount, the shape stays identical.
If $f(x) = x^2$ has its lowest point at $(0, 0)$, then $f(x) + 3 = x^2 + 3$ has the same U-shape but its lowest point is at $(0, 3)$.
The curve is simply lifted three units.

A vertical shift changes the $y$-intercept and any maximum or minimum value.
Shifting up by $3$ adds $3$ to the $y$-intercept and to the vertex's height.
So reading off a new minimum or intercept is just the old value plus $c$.

This is different from a horizontal shift, which changes $x$ inside the function.
Writing $f(x - 2)$ moves the graph sideways, while $f(x) + 2$ moves it up.
Adding outside the function affects height; changing what goes inside affects left-right position.

If $f(x) = x^2$, describe the graph of $f(x) + 5$.
Adding $5$ outside the function shifts the whole graph up by $5$.
So the vertex moves from $(0, 0)$ to $(0, 5)$, with the same U-shape.

The graph of $g(x)$ has a $y$-intercept of $4$. What is the $y$-intercept of $g(x) - 6$?
Subtracting $6$ lowers every point, including the intercept, by $6$.
So the new $y$-intercept is $4 - 6 = -2$.

Does $f(x) + 2$ shift the graph up or sideways?
The $2$ is added outside the function, so it changes the height, not the $x$-position.
So the graph shifts up by $2$.