Mathematics

Transforming functions

Turn one basic function into many: shift, flip, stretch, all with small moves.

What you need first

Once you know a basic function like $f(x) = x^2$ , you don't have to recompute everything for its relatives. You can take the finished graph and simply shift, flip, or stretch it. A few small moves are enough to turn one curve into a whole family.

Shifting up and down

If you add a number $c$ on the outside, that is $f(x) + c$ , the whole graph slides up by $c$ . If $c$ is negative, it goes down. Every point keeps its x-position, only its height changes. With $x^2 + 3$ it is the same as with $x^2$ , just three units higher.

f(x) + c \quad\Rightarrow\quad (x,\, y) \longrightarrow (x,\, y + c)

Adding on the outside shifts vertically: every point keeps its

x

and gains

c

in height.

Shifting left and right

If the number sits on the inside, at $f(x - d)$ , the graph moves horizontally. Careful, this feels backwards: $f(x - 2)$ moves 2 to the right, $f(x + 2)$ moves 2 to the left. The reason: with $f(x - 2)$ , $x$ has to grow by 2 before the bracket holds the same value as before. Every height is therefore reached 2 units later, and that is exactly what a shift to the right looks like.

Coordinate grid

y = 1x² + 1opens upwards

Move the sliders and watch the parabola travel and stretch without losing its shape.

Stretching and squashing

A factor on the outside, $a \cdot f(x)$ , pulls the graph taller. With $a = 2$ every height doubles, the curve rises twice as steeply, and that is exactly why a parabola looks narrower. With $0 < a < 1$ every height shrinks and the graph gets flatter. If $a$ is negative, the graph also flips over the x-axis: a valley becomes a hill.

g(x) = a \cdot f(x - d) + c

All together:

a

stretches the heights, the

d

in the bracket shifts the graph

d

to the right, and the

c

on the outside lifts it by

c

Exercises

0 of 6 solved

Time to try it yourself. You can't break anything, every attempt counts.

How do you get the graph of $f(x) + 4$ from the graph of $f(x)$ ?

In which direction does $f(x - 3)$ shift the graph?

Let $f(x) = x^2$ . What is the value of $f(x) + 5$ at $x = 2$ ?

Match each change to $f(x)$ with its effect on the graph.

f(x) - 2

f(x + 2)

-f(x)

2 \cdot f(x)

Let $f(x) = x^2$ . The graph is shifted 3 to the right, so $g(x) = (x - 3)^2$ . What is the value of $g$ at $x = 5$ ?

A negative $a$ in $a \cdot f(x)$ flips the graph over the … .

Where this leads

Inverse functions

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