Interactive Exploration of Functions

Welcome to this interactive guide to the fundamental families of functions. This module is designed to provide an intuitive, hands-on understanding of how different functions work and how their parameters alter their behavior. By manipulating the controls for each function, you can instantly visualize the concepts and build a strong foundation for more advanced topics in mathematics and computer science.

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Learning Objectives

  • Analyze and Construct Functions: Build linear and quadratic functions from geometric conditions and determine the domain and range of composite functions.
  • Interpret Function Parameters: Describe how changing parameters transforms the graph of linear, quadratic, and trigonometric functions.
  • Identify Key Features: Recognize zeros, degree, and asymptotes of polynomial and rational functions from their algebraic form.
  • Solve Equations: Find solutions for equations involving exponential, logarithmic, and trigonometric functions.
  • Relate Forms: Explain the relationship between a function and its inverse, both algebraically and graphically.

Linear Functions

A linear function represents a constant rate of change. Its graph is a straight line, defined by the equation y = mx + b, where 'm' is the slope and 'b' is the y-intercept. This section lets you manipulate these two parameters to see their direct effect on the line's position and steepness within a fixed viewing window.

Controls

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Quadratic Functions

A quadratic function forms a parabola. We use the vertex form y = a(x - h)² + k, where 'a' controls the parabola's width and direction, and '(h, k)' is the vertex. This is a very intuitive way to see how a parabola can be stretched, compressed, and shifted around the graph inside a consistent frame.

Controls

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Composite Functions

A composite function, or a "function of a function," is created when the output of one function becomes the input for another. Here, we explore h(x) = g(f(x)). Use the controls to define the quadratic functions f(x) and g(x), and choose an input value 'x'. The charts below visualize the process: the input 'x' produces an output 'f(x)' on the first graph. This output then becomes the input for the second graph, producing the final result 'g(f(x))'.

Master Control

Inner Function: f(x) = ax² + bx + c

Outer Function: g(x) = ax² + bx + c

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Inverse Functions

An inverse function reverses the action of another. If a function f(x) maps 'a' to 'b', its inverse f⁻¹(x) maps 'b' back to 'a'. Graphically, a function and its inverse are perfect reflections across the line y = x. Use the controls to define a linear function, then find its inverse and see the algebraic steps.

Function Controls

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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.

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Rational Functions

A rational function is a ratio of two polynomials. Their most interesting features are asymptotes: lines that the graph approaches but never touches. A vertical asymptote occurs where the denominator is zero, and a horizontal asymptote describes the function's behavior as x approaches infinity. Explore the function y = (ax²+bx+c)/(dx²+ex+f) below.

Controls

Numerator: ax² + bx + c

Denominator: dx² + ex + f

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Exponential & Logarithmic Functions

Exponential functions, like y = bˣ, model rapid growth or decay and are fundamental in computing and finance. Their inverse is the logarithmic function, y = logₐ(x). Manipulate the base 'b' to see how it affects the steepness of the growth curve and its corresponding logarithm, which is its reflection across y=x.

Controls

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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

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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.

Controls

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The Tangent Function

The tangent function is the ratio of sine to cosine. It is also periodic but has a range of all real numbers and vertical asymptotes where the cosine value is zero. You can manipulate its parameters just like sine and cosine to see how it stretches and shifts.

Controls

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Inverse Trigonometric Functions

To create inverse functions for periodic trigonometric functions, we must restrict their domains. For example, to find arcsin(x), we only consider the part of the sine wave from -π/2 to π/2. This ensures a unique output angle for each input value. The graph shows the original function, its restricted domain, and the resulting inverse reflected over y=x.

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