Starting from the graph of a known function, we can obtain the graphs of many related functions by applying simple geometric operations: shifting, reflecting, and scaling. Mastering these transformations eliminates the need to re-plot every function from scratch.
Quick Reference
| Transformation | Effect on graph | How each point $(a, b)$ moves |
|---|---|---|
| $y = f(x) + c$, $c > 0$ | Shift up $c$ units | $(a, b) \to (a, b+c)$ |
| $y = f(x) - c$, $c > 0$ | Shift down $c$ units | $(a, b) \to (a, b-c)$ |
| $y = f(x - c)$, $c > 0$ | Shift right $c$ units | $(a, b) \to (a+c, b)$ |
| $y = f(x + c)$, $c > 0$ | Shift left $c$ units | $(a, b) \to (a-c, b)$ |
| $y = -f(x)$ | Reflect in $x$-axis | $(a, b) \to (a, -b)$ |
| $y = f(-x)$ | Reflect in $y$-axis | $(a, b) \to (-a, b)$ |
| $y = cf(x)$, $c > 1$ | Vertical stretch by $c$ | $(a, b) \to (a, cb)$ |
| $y = cf(x)$, $0 < c < 1$ | Vertical compress by $c$ | $(a, b) \to (a, cb)$ |
| $y = f(cx)$, $c > 1$ | Horizontal compress by $1/c$ | $(a, b) \to (a/c, b)$ |
| $y = f(cx)$, $0 < c < 1$ | Horizontal stretch by $1/c$ | $(a, b) \to (a/c, b)$ |
| $y = |f(x)|$ | Reflect negative part up | Points below $x$-axis flip above |
| $y = f(|x|)$ | Reflect right side across $y$-axis | Left side replaced by mirror of right |
Vertical Shifts: $y = f(x) \pm c$
To obtain the graph of $y = f(x) + c$ ($c > 0$), shift the graph of $f$ upward by $c$ units.
To obtain the graph of $y = f(x) - c$ ($c > 0$), shift the graph of $f$ downward by $c$ units.
Horizontal Shifts: $y = f(x \mp c)$
To obtain the graph of $y = f(x - c)$ ($c > 0$), shift the graph of $f$ right by $c$ units.
To obtain the graph of $y = f(x + c)$ ($c > 0$), shift the graph of $f$ left by $c$ units.
Key reminder: subtracting from the input shifts the graph right; adding to the input shifts it left. This is the opposite of what you might expect.
Use the graph of $f(x) = x^2$ to sketch: (a) $g(x) = x^2 - 1$, (b) $h(x) = (x+2)^2$, (c) $F(x) = (x-2)^2$, (d) $G(x) = (x-2)^2 + 2$.
Solution
(a) $g(x) = f(x) - 1$: shift the parabola down 1 unit.
Reflection in the $x$-Axis: $y = -f(x)$
The graph of $y = -f(x)$ is obtained by reflecting the graph of $y = f(x)$ in the $x$-axis. Every point $(a, b)$ maps to $(a, -b)$.
Reflection in the $y$-Axis: $y = f(-x)$
The graph of $y = f(-x)$ is obtained by reflecting the graph of $y = f(x)$ in the $y$-axis. Every point $(a, b)$ maps to $(-a, b)$.
Vertical Scaling: $y = cf(x)$
- If $c > 1$: vertically stretch the graph of $f$ away from the $x$-axis by factor $c$.
- If $0 < c < 1$: vertically compress the graph toward the $x$-axis by factor $c$.
- If $c < 0$: apply the stretch or compression, then reflect in the $x$-axis.
Horizontal Scaling: $y = f(cx)$
- If $c > 1$: horizontally compress the graph toward the $y$-axis by factor $1/c$.
- If $0 < c < 1$: horizontally stretch the graph away from the $y$-axis by factor $1/c$.
- If $c < 0$: apply the stretch or compression, then reflect in the $y$-axis.
Combining Transformations: $y = kf(ax - b) + h$
To graph $y = kf(ax - b) + h$ from the graph of $y = f(x)$, apply transformations in this order:
- Horizontal scaling by $1/|a|$ (reflect in $y$-axis if $a < 0$).
- Horizontal shift by $b/a$ units.
- Vertical scaling by $|k|$ (reflect in $x$-axis if $k < 0$).
- Vertical shift by $h$ units.
Use the graph of $y = \sqrt{x}$ to obtain the graph of $y = 2 - \sqrt{2x + 4}$ and determine its domain and range.
Solution
Rewrite: $y = -\sqrt{2(x+2)} + 2$. Step 1: $y = \sqrt{2x}$ — horizontal compression of $y = \sqrt{x}$ by factor $1/2$.
Special Transformations: $y = |f(x)|$ and $y = f(|x|)$
$y = |f(x)|$: Reflect any part of the graph of $f$ that lies below the $x$-axis up to above the $x$-axis. The part already above stays unchanged.
$y = f(|x|)$: Remove the part of the graph for $x < 0$, keep the right half ($x \geq 0$), and reflect it in the $y$-axis to fill the left side. The result is always an even function.
