Activities to Teach Students to Determine End Behavior Using Graphs
When teaching students about the behavior of a function, one important concept to understand is the end behavior, which describes how the function behaves as x approaches positive or negative infinity. Understanding end behavior can help students evaluate limits, sketch graphs, and determine important properties of functions.
Here are some activities that can help students practice determining end behavior using graphs:
1. Use a visual approach: One way to help students understand end behavior is to use a visual approach, such as a graphing calculator or online graphing tool. Begin by graphing a function and zooming out to show how the curve behaves as x approaches infinity or negative infinity. For example, if the function is f(x) = x^2 – 4x + 3, graph the function and observe how the curve approaches the x-axis as x becomes very large (for positive infinity) or very negative (for negative infinity).
2. Analyze the leading coefficient: Another way to help students determine end behavior is to analyze the leading coefficient of the function. For example, if the function is f(x) = -2x^3 + 4x^2 – 5x + 2, the leading coefficient is -2. If the leading coefficient is positive, the end behavior will be similar to a positive quadratic (upward) or cubic (with both ends going up).
3. Use substitution: One method to find end behavior of a polynomial function is to substitute large or small values of x into the function to see what happens to the output. When x is very large and positive, for example, substitute x = 1000 into the function and observe the resulting output. Then, substitute x = -1000 and observe the output. This can help students determine whether the function approaches a positive or negative infinity at the end points.
4. Compare with known functions: Another way to determine end behavior is to compare the given function to known functions. For example, if the function is f(x) = 1/x, analyze how the graph behaves as x approaches infinity or negative infinity, and compare it to the graph of y = 1/x^2. By comparing the two functions, students can determine the end behavior of the original function.
By using these activities, teachers can help students develop a deeper understanding of end behavior, which is essential for understanding limits and properties of functions. With practice, students can become more confident in analyzing graphs and evaluating limits, which will help them in future math courses and real-world situations.