#businessarticle8 #mathematics #linearandquadraticfunctions

Rules for linear and quadratic functions

By Lona Matshingana 

2026/01/01

# Linear and Quadratic Functions in Pure Mathematics

## Linear Functions

A **linear function** has the form:

$$f(x) = mx + b$$

where $m$ and $b$ are real constants.

**Key Properties:**

- **Degree**: 1 (the highest power of $x$ is 1)

- **Graph**: A straight line in the coordinate plane

- **Slope**: $m$ represents the rate of change (constant)

- **y-intercept**: $b$ is where the line crosses the y-axis

- **Domain and Range**: Both are all real numbers (unless restricted)

- **Continuity**: Continuous everywhere

- **Monotonicity**: Strictly increasing if $m > 0$, strictly decreasing if $m < 0$, constant if $m = 0$

**Special cases:**

- If $m = 0$, the function is constant: $f(x) = b$

- If $b = 0$, the function passes through the origin: $f(x) = mx$

## Quadratic Functions

A **quadratic function** has the form:

$$f(x) = ax^2 + bx + c$$

where $a, b, c$ are real constants and $a \neq 0$.

**Key Properties:**

- **Degree**: 2 (the highest power of $x$ is 2)

- **Graph**: A parabola

- **Vertex**: The turning point at $x = -\frac{b}{2a}$, giving vertex $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

- **Axis of symmetry**: The vertical line $x = -\frac{b}{2a}$

- **Direction**: Opens upward if $a > 0$, downward if $a < 0$

- **y-intercept**: $c$ (the value when $x = 0$)

- **Domain**: All real numbers

- **Range**: $[k, \infty)$ if $a > 0$ or $(-\infty, k]$ if $a < 0$, where $k$ is the y-coordinate of the vertex

- **Continuity**: Continuous everywhere

- **Extremum**: Has a minimum if $a > 0$ or maximum if $a < 0$ at the vertex

**Roots (zeros):**

Found using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The **discriminant** $\Delta = b^2 - 4ac$ determines the nature of roots:

- $\Delta > 0$: Two distinct real roots

- $\Delta = 0$: One repeated real root (the parabola touches the x-axis)

- $\Delta < 0$: No real roots (two complex conjugate roots)

**Vertex form:**

$$f(x) = a(x - h)^2 + k$$

where $(h, k)$ is the vertex.

**Factored form** (when roots $r_1$ and $r_2$ exist):

$$f(x) = a(x - r_1)(x - r_2)$$

These forms are all equivalent and useful for different purposes in analysis and problem-solving.

Thank you for reading!!!