Move y = x and y = x^2 around the plane by editing their equations: add outside to shift vertically, replace x inside to shift horizontally with the sign reversed, and watch the bracket get squared whole.
Want to move a graph without drawing it again? Just change its equation. Take two parent functions, and , and watch what moves them up, down, left, or right.
To move a function upward, add a number to it. To move it downward, subtract a number. The graph's shape never changes — only its position.
A positive number lifts the graph; a negative one lowers it. The sign matches the direction — the intuitive half of the lesson.
This half does not behave the way you'd guess. Instead of adding to the whole function, replace every inside the equation — and the sign runs backwards.
Apply the horizontal rule to . Every is replaced — including the one being squared.
2 units right
5 units left
The vertex started on the y-axis. After the second transformation it sits 5 units to the left of it — watch the vertex to check any shift.
For , the function produces its original output when , that is when . So whatever used to happen at now happens at — the whole graph has slid to meet it, two units to the right.