Common Core Algebra II.Unit 10.Lesson 1.Power Functions

Introduction

This tutorial covers the concepts of power functions as presented in Common Core Algebra II, Unit 10, Lesson 1. Understanding power functions is essential for solving various mathematical problems and can be applied in real-world situations, such as physics and engineering. In this lesson, we will break down the key features of power functions, their graphs, and how to analyze them.

Step 1: Understand Power Functions

Power functions are mathematical expressions of the form:

f(x) = k * x^n

Where:

• k is a constant (the coefficient).
• x is the variable.
• n is the exponent (which can be positive, negative, or zero).

Key Points:

• If n is a positive integer, the graph will generally rise to the right.
• If n is negative, the graph will approach the x-axis as x increases.
• If n is zero, the function simplifies to a constant (f(x) = k).

Step 2: Identify Characteristics of Power Functions

Power functions have specific characteristics based on the value of n.

Characteristics:

1. Even vs. Odd Exponents:

   • Even Exponents (e.g., n = 2): The graph is symmetric about the y-axis.
   • Odd Exponents (e.g., n = 3): The graph is symmetric about the origin.

2. End Behavior:

   • For even n: As x approaches ±∞, f(x) approaches +∞.
   • For odd n: As x approaches +∞, f(x) approaches +∞, and as x approaches -∞, f(x) approaches -∞.

3. Intercepts:

   • The y-intercept occurs at (0, k), provided k is not zero.
   • The x-intercepts depend on the equation set to zero.

Step 3: Graphing Power Functions

To graph a power function, follow these steps:

1. Determine Key Values:

   • Calculate f(x) for several values of x (both positive and negative).
   • Choose values close to zero to see the behavior near the y-axis.

2. Plot Points:

   • Plot the calculated points on a coordinate plane.

3. Draw the Curve:

   • Connect the points smoothly, considering the characteristics of the function (even or odd).

Practical Tip:

Use graphing software or a calculator for complex functions to visualize the graphs accurately.

Step 4: Analyze Power Functions

To analyze power functions:

1. Find the Domain and Range:

   • Domain: Generally all real numbers, but check for restrictions.
   • Range: Depends on the value of k and n.

2. Evaluate Specific Points:

   • Substitute specific x values into the function to determine f(x).

3. Check Asymptotic Behavior:

   • Analyze the behavior of the function as x approaches infinity or negative infinity.

Conclusion

Power functions are a fundamental concept in algebra that can be applied to various mathematical problems. By understanding their structure, characteristics, and how to graph them, you can better analyze and interpret these functions. As a next step, practice graphing different power functions and exploring their properties to reinforce your understanding.