Up and Down

Part of the Functions & Transformations section of Coddy's Algebra 2 journey. Lesson 30 of 55.

This chapter moves graphs around without changing their shape. Every example starts from one parent, $f\left(x\right) = {x}^{2}$ . Its lowest point, the vertex, is at (0, 0), and one step to the right the curve has climbed to (1, 1).

Take $y = f\left(x\right)+2$ . For each input it works out $f\left(x\right)$ and then adds 2, so every output is 2 more than before. The vertex (0, 0) rises to (0, 2) and the point (1, 1) rises to (1, 3). Every point climbs the same 2, so the whole curve moves up 2 and keeps its shape.

$The parabola y = x^2 dashed, and the same curve moved up 2 with its vertex at (0, 2)$

The dashed parent, and the same curve lifted 2.

A number added outside the function changes heights only. Adding k moves the graph up k, and subtracting moves it down: $y = f\left(x\right)-5$ puts the vertex at (0, -5).

Challenge

Draw the graph of $y = f\left(x\right)-3$ for $f\left(x\right) = {x}^{2}$ . The faint curve is f itself. Drag the vertex, then the second point, until your curve matches.