A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The graph is a straight line with constant rate of change.
$f(x) = 2x - 3$. Slope $m = 2$, y-intercept $b = -3$. Domain: all real numbers $(-\infty, \infty)$. Range: all real numbers $(-\infty, \infty)$.
A quadratic function has the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph is a parabola that opens upward if $a > 0$ and downward if $a < 0$.
$f(x) = x^2 - 4x + 3$. Complete the square: $f(x) = (x-2)^2 - 1$. Vertex: $(2, -1)$. Domain: $(-\infty, \infty)$. Range: $[-1, \infty)$.
A polynomial function has the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$, where $n$ is a non-negative integer. The degree of the polynomial is the highest power $n$.
| Degree | Type | Example | Shape |
|---|---|---|---|
| 0 | Constant | $f(x) = 5$ | Horizontal line |
| 1 | Linear | $f(x) = 2x + 1$ | Straight line |
| 2 | Quadratic | $f(x) = x^2$ | Parabola |
| 3 | Cubic | $f(x) = x^3$ | S-shaped |
A rational function is the ratio of two polynomials: $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$. The domain excludes values where $Q(x) = 0$.
Find the domain of $f(x) = \frac{2x}{x^2 - 4}$.
A radical function contains a root, such as $\sqrt{x}$ or $\sqrt[3]{x}$. For square roots (even roots), the radicand must be non-negative. For cube roots (odd roots), the domain is all real numbers.
| Function | Domain | Range |
|---|---|---|
| $f(x) = \sqrt{x}$ | $[0, \infty)$ | $[0, \infty)$ |
| $f(x) = \sqrt{x-2}$ | $[2, \infty)$ | $[0, \infty)$ |
| $f(x) = \sqrt[3]{x}$ | $(-\infty, \infty)$ | $(-\infty, \infty)$ |
Find the domain of $f(x) = \sqrt{5 - x}$.
Solution: $5 - x \ge 0 \Rightarrow x \le 5$. Domain: $(-\infty, 5]$.
The absolute value function is $f(x) = |x|$, defined as:
The graph is V-shaped, with vertex at $(0,0)$. Domain: all real numbers. Range: $[0, \infty)$.
Graph $f(x) = |x-2| + 1$. The vertex shifts to $(2, 1)$. Domain: $(-\infty, \infty)$. Range: $[1, \infty)$.
Interactive graph below. Click buttons to see different function families:
Find the slope and y-intercept of $f(x) = -3x + 5$.
Slope $m = -3$, y-intercept $b = 5$.
Find the vertex of $f(x) = x^2 + 6x + 8$.
Complete the square: $x^2 + 6x + 8 = (x+3)^2 - 1$. Vertex: $(-3, -1)$.
Find the domain of $f(x) = \frac{x+1}{x^2 - 9}$.
Denominator: $x^2 - 9 = (x-3)(x+3) = 0$ when $x = 3$ or $x = -3$. Domain: $(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$.
Find the domain of $f(x) = \sqrt{2x + 6}$.
$2x + 6 \ge 0 \Rightarrow 2x \ge -6 \Rightarrow x \ge -3$. Domain: $[-3, \infty)$.