๐ Topic Summary
End behavior describes the trend of the graph of a function as $x$ approaches positive infinity ($+\infty$) and negative infinity ($-\infty$). In simpler terms, it tells us what direction the graph is going on the far left and the far right. For polynomial functions, the end behavior is determined by the leading term (the term with the highest power of $x$). The degree (highest power) and the leading coefficient (the number multiplied by the highest power of $x$) dictate whether the graph rises or falls on each end.
Understanding end behavior is crucial for sketching polynomial graphs and analyzing the behavior of functions in various mathematical models. By knowing the end behavior, we can get a general idea of the shape of the graph even without plotting numerous points. It's all about looking at the big picture! ๐
๐ค Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Leading Coefficient |
A. The behavior of the graph as $x$ approaches $+\infty$ or $-\infty$ |
| 2. Degree |
B. The term in a polynomial with the highest power of the variable. |
| 3. End Behavior |
C. The highest power of the variable in a polynomial. |
| 4. Leading Term |
D. The number multiplied by the variable with the highest power. |
| 5. Polynomial Function |
E. A function that can be written in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ |
โ๏ธ Part B: Fill in the Blanks
Complete the following paragraph using the words: degree, leading coefficient, positive, negative, end behavior.
The _______ of a polynomial function and the _______ determine its _______. If the leading coefficient is _______ and the degree is even, the graph rises on both ends. If the leading coefficient is _______ and the degree is even, the graph falls on both ends.
๐ค Part C: Critical Thinking
Explain how the end behavior of a polynomial function can help you identify potential errors when graphing it. Provide a specific example.