Power functions are functions in the form $f(x) = kx^p$, where $k$ is a constant and $p$ is a real number. The shape of the graph depends heavily on the value of $p$. If $p$ is an even integer, the graph resembles a parabola. If $p$ is an odd integer, the graph resembles a cubic function. When $p$ is a fraction, the graph will have different characteristics, especially concerning the domain and range. Understanding the effect of $p$ on the graph is key to sketching and analyzing power functions.
This quiz will test your knowledge of identifying key features, matching equations to graphs, and understanding how transformations affect power functions. Good luck!
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Power Function | A. The point where the graph crosses the x-axis. |
| 2. Exponent | B. A function of the form $f(x) = kx^p$, where $k$ is a constant and $p$ is a real number. |
| 3. Coefficient | C. The value that is raised to a power. |
| 4. Root | D. A number multiplied by a variable. |
| 5. Base | E. A symbol or number indicating the power to which a base is raised. |
Complete the following paragraph with the correct terms:
The general form of a power function is $f(x) = kx^p$, where $k$ is the ________ and $p$ is the ________. When $p$ is a positive ________, the graph typically passes through the origin. The ________ of the function is affected by the value of $p$, determining its increasing or decreasing behavior. If $k$ is negative, the graph is ________ across the x-axis.
Explain how changing the value of $k$ in the power function $f(x) = kx^p$ affects the graph. Provide examples using different values of $k$ and describe the transformations that occur.
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