What is a Function?
In mathematics, a function is a rule that assigns to each input exactly one output. Think of it like a machine: you put something in, and you get something out. For example, a function might take a number as input and output its square. If you input 2, the output is 4. If you input 3, the output is 9. The key idea is that for any given input, you always get the same output.
We can represent functions in several ways:
- Verbally: "The output is the input multiplied by two."
- Algebraically: $f(x) = 2x$
- Numerically: A table of values.
- Graphically: A plot on a coordinate plane.
Functions are the building blocks of mathematics and are used to model real-world phenomena, from the trajectory of a thrown ball to the growth of a population.
Linear Functions: The Straight and Narrow
The simplest type of function, a linear function, creates a straight line when graphed. It has the form $f(x) = mx + b$.
Equation: y = 1x + 0
Quadratic Functions: The Elegant Curve
A quadratic function, in the form $f(x) = ax^2 + bx + c$, creates a U-shaped curve called a parabola.
Equation: y = 1x² + 0x + 0
Polynomial Functions
Any set of points can be described by a unique polynomial function. Click on the chart below to add points. Then, use the controls to find the polynomial that passes through them. This process is known as polynomial interpolation.
Interpolation Details
Click on the chart to add points.
Exponential Functions: Rapid Growth and Decay
Exponential functions, of the form $f(x) = a \cdot b^x$, model phenomena that grow or shrink at a rate proportional to their current size.
Equation: y = 1 * 2^x
The Unit Circle and Trigonometric Waves
The Unit Circle is a powerful tool for understanding trigonometry. Hover your mouse over the circle to see how the angle (θ) corresponds to the x and y values on the sine and cosine waves. Press "Play" to watch the point revolve around the circle and trace the waves in real time.
Angle (rad): 0.00
Angle (deg): 0.0°
cos(θ) = 1.000
sin(θ) = 0.000
Sine & Cosine Functions
Sine and Cosine are periodic functions that model wave-like phenomena. The equation y = A·sin(B(x-C)) + D allows us to transform the basic wave. 'A' is amplitude, 'B' is frequency, 'C' is phase shift, and 'D' is vertical shift. A toggle is provided to switch between the Sine and Cosine functions.