Algebra & Functions: Linear, Quadratic & Exponential
Algebra and functions form the backbone of mathematics from high school onwards. Whether you are graphing a straight-line budget, solving for when a projectile hits the ground, or modelling population growth, the same core ideas about functions recur again and again. This guide explains linear, quadratic, and exponential functions in practical terms, shows you the key formulas, and points you to tools that let you explore any equation visually.
What Is a Function?
A function is a rule that assigns exactly one output value to every input value. We write this as f(x), read “f of x,” where x is the input (the independent variable) and the result is the output (the dependent variable). Plotting the pairs (x, f(x)) on a coordinate grid gives the function's graph, which makes its behaviour much easier to understand than a table of numbers alone.
Linear Functions: y = mx + b
A linear function produces a straight line. Its equation is y = mx + b, where:
- m is the slope — how steeply the line rises or falls (rise ÷ run between any two points).
- b is the y-intercept — the value of y when x = 0, i.e., where the line crosses the vertical axis.
A positive slope means the line goes up left-to-right; a negative slope goes down. A slope of zero gives a flat horizontal line. The Linear Function Plotter lets you enter any m and b and see the resulting line immediately, which is especially useful when experimenting with multiple scenarios.
To find the slope between two known points (x₁, y₁) and (x₂, y₂), use the Slope Calculator: m = (y₂ − y₁) ÷ (x₂ − x₁).
Quadratic Functions: ax² + bx + c
A quadratic function is a polynomial of degree 2. Its equation is y = ax² + bx + c, and its graph is a U-shaped (or inverted-U) curve called a parabola. The coefficient a determines the direction and width of the parabola: a positive a opens upward (a bowl), a negative a opens downward (a dome).
Key features of a parabola:
- Vertex: the highest or lowest point, at x = −b ÷ (2a).
- Axis of symmetry: the vertical line through the vertex, x = −b ÷ (2a).
- Roots (x-intercepts): where y = 0, found using the quadratic formula.
Use the Quadratic Function Plotter to visualise any parabola by entering your a, b, and c values. To find the exact roots (real or complex), use the Quadratic Equation Solver, which applies the quadratic formula x = (−b ± √(b² − 4ac)) ÷ (2a).
The Quadratic Formula: Worked Example
Solve x² − 5x + 6 = 0. Here a = 1, b = −5, c = 6.
- Discriminant: b² − 4ac = 25 − 24 = 1
- x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2
- x = 3 or x = 2
When the discriminant is positive, there are two real roots. When it is zero, there is exactly one root. When it is negative, there are no real roots (only complex ones).
Exponential Functions: Growth and Decay
Exponential functions model situations where the rate of change is proportional to the current value — for example, compound interest, bacterial growth, radioactive decay, or the cooling of an object.
| Type | Formula | Behaviour |
|---|---|---|
| Exponential growth | y = a × (1 + r)t | Value rises faster and faster over time |
| Exponential decay | y = a × (1 − r)t | Value falls quickly at first, then more slowly |
Where a is the starting value, r is the rate (as a decimal), and t is time. The Exponential Growth Calculator and the Exponential Decay Calculator handle both scenarios, making them useful for science coursework, financial projections, and ecology problems.
Worked Example: Compound Interest as Exponential Growth
You invest $1,000 at 6% annual interest, compounded yearly. How much do you have after 10 years?
y = 1000 × (1 + 0.06)10 = 1000 × 1.7908 = $1,790.85
The characteristic “hockey stick” shape of exponential growth means the gains accelerate — more than half the total growth occurs in the final three years of that ten-year window.
Common Mistakes
- Confusing slope with y-intercept. In y = mx + b, m is always the multiplier of x, and b is always the standalone constant.
- Forgetting the ± in the quadratic formula. Both solutions must be checked — one or both may not satisfy the original problem's constraints.
- Using percentage rates without converting to decimals. In exponential formulas, 6% must be entered as 0.06, not 6.
Frequently Asked Questions
How do I know if a function is linear, quadratic, or exponential?
Check the equation: a linear function has x to the first power only; a quadratic has x² as its highest term; an exponential has x in the exponent (e.g., 2x or ex). On a graph, a straight line is linear, a symmetric curve is quadratic, and a rapidly accelerating or decelerating curve is exponential.
What does a negative discriminant mean?
It means the quadratic has no real roots — the parabola does not cross the x-axis. The solutions exist as complex (imaginary) numbers.
Can exponential growth continue forever?
In theory, yes mathematically. In practice, real-world systems always encounter limiting factors — finite resources, carrying capacity, market saturation — that cause growth to slow and often follow a logistic S-curve instead.