Graph Transformations

Starting point: Years 9 and 10

Start with $y=x^2$, vertex at the origin.

• $y=x^2+3$ moves it up 3.
• $y=x^2-3$ moves it down 3.
• $y=(x-2{)}^2$ moves it right 2.
• $y=(x+2{)}^2$ moves it left 2.
• $y=-x^2$ flips it upside down.
• $y=3x^2$ makes it narrower (steeper). $y=\frac{1}{3}{x}^2$ makes it wider.

Quick summary

• $f(x)+c$ moves up $c$. $f(x-b)$ moves right $b$.
• $-f(x)$ flips in the x-axis. $f(-x)$ flips in the y-axis.
• $kf(x)$ stretches vertically by $k$. $f(ax)$ stretches horizontally by $\frac{1}{a}$.
• Changes inside the bracket act on $x$ and do the opposite of what they look like.
• For $y=kf(a(x+b))+c$, factor out $a$ and work from the inside out.

Quiz