When comparing linear, quadratic, cubic, and exponential functions, one key feature that all four types of functions share is that they can all be expressed as mathematical equations. Each function is defined by a specific formula that describes the relationship between the input (usually denoted as \(x\)) and the output (usually denoted as \(f(x)\)).
In more detail:
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Linear Functions: These are of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. They produce straight lines on a graph.
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Quadratic Functions: These have the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. They produce parabolas.
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Cubic Functions: These are expressed as \(f(x) = ax^3 + bx^2 + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are constants. They produce curves that can have an "S" shape.
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Exponential Functions: These are of the form \(f(x) = a \cdot b^x\), where \(a\) is a constant and \(b\) is a positive constant base. They exhibit rapid growth or decay, depending on the base.
Moreover, all four function types can be graphed, analyzed for their properties (such as increasing/decreasing behavior, intercepts, etc.), and can be used to model real-world situations.
Another common feature is that they can all be categorized based on their growth rates, what differentiates them is how they grow—linear grows at a constant rate, quadratic grows at an increasing rate (accelerating), cubic grows even faster, and exponential grows at a much faster rate than any polynomial as \(x\) increases.
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