The area, A, of a rectangle whose length is 3 times its...

The area, $A,$ of a rectangle whose length is 3 times its width is given by $A=3 w^{2}$, where $w$ is its width.
(a) Identify the coefficient and exponent of this power function.
(b) If the width is $5 \mathrm{~cm}$, what is the area of the rectangle?

Key Concepts

Power Function

A power function is a mathematical expression of the form f(x) = kx^n, where k is a constant coefficient and n is a constant exponent. It represents a type of relationship where the output of the function depends on raising the input to a specified power, thereby modeling various real-world phenomena such as growth, decay, and scaling in geometric shapes.

Coefficient

In a power function, the coefficient is the constant factor that multiplies the variable raised to the power. It determines the scale or magnitude of the function's output and can affect the steepness or rate of change, thereby directly influencing how the function behaves as the input changes.

Exponent

The exponent in a power function indicates the power to which the input variable is raised. It plays a crucial role in determining the nature of the function, such as whether it represents a linear, quadratic, or higher-degree relationship. The exponent shapes the curve of the function, affecting how rapidly the output increases or decreases with changes in the variable.

Function Evaluation

Function evaluation is the process of substituting a specific value into a mathematical function to compute the output. This technique is fundamental for determining the value of the function at a given point and is widely used in various areas of mathematics to analyze relationships and solve problems.

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