Algebra 2 / Unit 1: Functions and Transformations
1.2 Transformations of Functions
Notes
The form $g(x) = a\,f(x - h) + k$ moves and reshapes the graph of $f$.
- $h$ shifts horizontally: $f(x - 2)$ moves right 2, $f(x + 2)$ moves left 2. The sign feels backward.
- $k$ shifts vertically: $+k$ moves up, $-k$ moves down.
- $|a| > 1$ is a vertical stretch, $0 < |a| < 1$ is a vertical shrink, and $a < 0$ reflects over the $x$-axis.
- $f(-x)$ reflects over the $y$-axis.
- Example: a point $(x, y)$ on $f$ moves to $(x + h, ay + k)$ on $g$.
Practice
Compared with $y = x^2$, the graph of $g(x) = (x + 5)^2 - 2$ is shifted:
$x + 5 = x - (-5)$, so $h = -5$ (left), and $k = -2$ (down).
The graph of $g(x) = -f(x)$ is the graph of $f$:
Multiplying the output by $-1$ flips every $y$-value, a reflection over the $x$-axis.
The point $(2, 5)$ is on the graph of $f$. Which point must be on $g(x) = f(x - 3) + 1$?
Right 3 and up 1: $(2 + 3, 5 + 1)$.
Ready for more? Take the Unit 1 test.