- Identify power functions and its end behavior.
- Identify polynomial functions and its degree and leading coefficient.
- Identifying local behavior of polynomial functions.
Power Functions
Have you ever noticed how a small increase in the dimensions of an object can lead to a much larger increase in its volume? For example, doubling the side length of a cube results in a volume that’s eight times larger!
This fascinating relationship is due to power functions, a special type of mathematical function that can model a variety of real-world phenomena, from the growth of populations to the way light diminishes over distance. A power function is a function with a single term that is the product of a real number, coefficient, and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a coefficient.
power function
A power function is a function that can be represented in the form
[latex]f\left(x\right)=a{x}^{n}[/latex]
where [latex]a[/latex] and [latex]n[/latex] are real numbers and [latex]a[/latex] is known as the coefficient.
[latex]\begin{array}{c}f\left(x\right)=1\hfill & \text{Constant function}\hfill \\ f\left(x\right)=x\hfill & \text{Identity function}\hfill \\ f\left(x\right)={x}^{2}\hfill & \text{Quadratic}\text{ }\text{ function}\hfill \\ f\left(x\right)={x}^{3}\hfill & \text{Cubic function}\hfill \\ f\left(x\right)=\frac{1}{x} \hfill & \text{Reciprocal function}\hfill \\ f\left(x\right)=\frac{1}{{x}^{2}}\hfill & \text{Reciprocal squared function}\hfill \\ f\left(x\right)=\sqrt{x}\hfill & \text{Square root function}\hfill \\ f\left(x\right)=\sqrt[3]{x}\hfill & \text{Cube root function}\hfill \end{array}[/latex]
No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.