Precalculus / Unit 1: Functions and Their Graphs
1.2 Transformations of Graphs
Notes
Transformations shift, stretch or flip the graph of a parent function $f(x)$.
- $f(x - h)$ shifts right $h$; $f(x + h)$ shifts left $h$. $f(x) + k$ shifts up $k$.
- $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.
- $a f(x)$ stretches vertically by $|a|$; $f(bx)$ compresses horizontally by a factor of $\displaystyle \frac{1}{|b|}$.
- Changes inside the parentheses act on x and work "backwards" from what you might expect.
- Example: the vertex of $y = (x + 3)^2 - 2$ is $(-3, -2)$.
Practice
The graph of $g(x) = f(x - 4) + 1$ is the graph of $f$ shifted:
$x - 4$ moves right 4 and $+1$ outside moves up 1.
The point $(2, 6)$ is on $y = f(x)$. Which point is on $y = 2f(x + 1) - 3$?
Set $x + 1 = 2$ so $x = 1$. Then $y = 2(6) - 3 = 9$.
The point $(3, 5)$ is on $y = f(x)$. Which point is on $y = f(2x)$?
Need $2x = 3$, so $\displaystyle x = \frac{3}{2}$; the y-value stays 5.
Ready for more? Take the Unit 1 test.