Mathematics
Grade 12
15 min
Determine end behavior using graphs
Determine end behavior using graphs
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1
Introduction & Learning Objectives
Learning Objectives
Interpret the end behavior of a function by analyzing its graph.
Use formal limit notation to describe the end behavior of a function as x approaches positive and negative infinity.
Identify the equations of horizontal asymptotes by visually inspecting a function's end behavior.
Distinguish between end behavior that approaches a finite limit and end behavior that approaches positive or negative infinity.
Sketch the ends of a possible graph of a function given its end behavior in limit notation.
Connect the graphical concept of end behavior to the long-term trends of real-world models.
Ever wonder what happens to a population of bacteria in a petri dish over a very long time? 📈 The answer lies in understanding a graph's 'end game'!
Th...
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Key Concepts & Vocabulary
TermDefinitionExample
End BehaviorThe behavior of the y-values (f(x)) of a function as the x-values approach positive infinity (x → ∞) or negative infinity (x → -∞). It describes what happens at the extreme ends of the graph.For the graph of y = x², as x goes to the far right (∞), y also goes up to ∞. As x goes to the far left (-∞), y still goes up to ∞.
Limit at InfinityThe value that a function's output (y-value) approaches as the input (x-value) grows infinitely large in either the positive or negative direction.For the function f(x) = 1/x, the limit as x approaches infinity is 0, because the y-values get closer and closer to 0.
Horizontal AsymptoteA horizontal line, y = L, that the graph of a function approaches as x → ∞ or x → -∞. The existence of a horizontal asymptote is a dir...
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Core Formulas
Right-End Behavior
lim_{x \to \infty} f(x) = L
This notation describes the behavior of the graph as you trace it to the far right. The value 'L' can be a finite number (indicating a horizontal asymptote) or ∞ or -∞ (indicating unbounded behavior).
Left-End Behavior
lim_{x \to -\infty} f(x) = M
This notation describes the behavior of the graph as you trace it to the far left. The value 'M' can be a finite number or ∞ or -∞. Note that M and L are not necessarily the same.
Horizontal Asymptote from Limits
If lim_{x \to \infty} f(x) = L or lim_{x \to -\infty} f(x) = L (where L is a finite real number), then the line y = L is a horizontal asymptote.
This rule directly connects the concept of a finite limit at infinity to the geometric feature of a hori...
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Challenging
A graph of a function f(x) is shown. It has the end behavior lim_{x \to \infty} f(x) = 2 and lim_{x \to -\infty} f(x) = \infty. Consider the new function g(x) = 1/f(x). What is the end behavior of g(x)?
A.lim_{x \to \infty} g(x) = 1/2 and lim_{x \to -\infty} g(x) = 0
B.lim_{x \to \infty} g(x) = 2 and lim_{x \to -\infty} g(x) = \infty
C.lim_{x \to \infty} g(x) = -2 and lim_{x \to -\infty} g(x) = 0
D.lim_{x \to \infty} g(x) = 1/2 and lim_{x \to -\infty} g(x) = \infty
Challenging
The graph of a function f(x) is shown. It appears to approach a slant asymptote, the line y = x + 1, as x → ±∞. Which of the following limit statements best describes the end behavior?
A.lim_{x \to \infty} f(x) = 1 and lim_{x \to -\infty} f(x) = 1
B.lim_{x \to \infty} f(x) = \infty and lim_{x \to -\infty} f(x) = -\infty
C.lim_{x \to \infty} [f(x) - (x+1)] = 0
D.The limits do not exist.
Challenging
The graph of f(x) has the end behavior lim_{x \to \infty} f(x) = 5 and lim_{x \to -\infty} f(x) = -\infty. What is the end behavior of the transformed function g(x) = f(-x)?
A.lim_{x \to \infty} g(x) = 5 and lim_{x \to -\infty} g(x) = -\infty
B.lim_{x \to \infty} g(x) = -\infty and lim_{x \to -\infty} g(x) = 5
C.lim_{x \to \infty} g(x) = -5 and lim_{x \to -\infty} g(x) = \infty
D.lim_{x \to \infty} g(x) = 1/5 and lim_{x \to -\infty} g(x) = 0
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