Impact of the constant on the graph

A constant added at the end of a function shifts its graph straight up or down.
Adding moves it up; subtracting moves it down.
Every point moves by the same amount, so the shape stays the same.

The figure shows a parabola shifted by a constant.
Adding $3$ moves every point up by $3$, and subtracting $3$ moves every point down by $3$.
The curve keeps its exact shape — only its height changes.

In vertex form $y = a(x - h)^2 + k$, the constant $k$ is the $y$-value of the vertex, so it controls the up-down position.
For a line $y = mx + b$, the constant $b$ is the $y$-intercept; shifting the line down by $4$ changes $b$ to $b - 4$, while the slope stays the same.

A constant added at the end does not change the shape or steepness of the graph.
It also does not decide whether the function is increasing or decreasing.
It only slides the graph up or down.

The graph of $y = h(x)$ is shifted down $10$ units. What is the new function?
Shifting down by $10$ means subtracting $10$ from the whole function.
So the new function is $h(x) - 10$.

The line $y = 2x + 10$ is shifted down $4$ units. What is the new equation?
Subtract $4$ from the whole function: $y = 2x + 10 - 4$.
So $y = 2x + 6$, with the same slope and a lower $y$-intercept.

Parabola $g$ has the same shape as $f$, but its vertex sits $3$ units higher. How are their equations related?
Shifting up $3$ units means adding $3$ to the function.
So $g(x) = f(x) + 3$.