Asymptotes and graph behaviour
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Question
Define a horizontal asymptote.
Answer
A horizontal line y = k that the graph approaches as x โ โ or x โ โโ, but (usually) never reaches or crosses.
Question
Which function family always has a horizontal asymptote at y = 0 (if not vertically shifted)?
Answer
Exponential: y = a ยท bหฃ. As x โ โโ (for b > 1) or x โ โ (for 0 < b < 1), the output approaches 0.
Question
IB asks "Write down the equation of the horizontal asymptote." What is the required format?
Answer
Write it as a full equation: e.g. y = 3. Not just "3" โ the y = must be included.
Question
In plain language, what does "approaching an asymptote" mean?
Answer
As x gets very large (or very negative), the output of f gets arbitrarily close to the asymptote value โ but the curve never quite touches that line.
Question
State the horizontal asymptote of f(x) = 3 ยท 2หฃ + 5.
Answer
y = 5. As x โ โโ, 3 ยท 2หฃ โ 0, so f(x) โ 5. The +5 shifts the asymptote up from y = 0 to y = 5.
Question
How does the horizontal asymptote affect the range of f(x) = 2 ยท 3หฃ + 4?
Answer
Range is f(x) > 4. The function always stays above y = 4 (never equals it), so 4 is excluded from the range.
Question
f(x) = 100 ยท 0.5หฃ + 10. What is the horizontal asymptote and what happens as x โ โ?
Answer
Horizontal asymptote y = 10. As x โ โ, 100 ยท 0.5หฃ โ 0, so f(x) โ 10 from above.
Question
What does a horizontal asymptote tell you about the range of the function?
Answer
The function never reaches the asymptote value, so that value is excluded from the range. E.g. if asymptote y = 3 and function approaches from above, range is f(x) > 3.
Question
What is a vertical asymptote?
Answer
A vertical line x = a where the function is undefined and its output grows to ยฑโ as x approaches a from either side.
Question
Where does y = 1/(x โ 3) have a vertical asymptote?
Answer
At x = 3 โ the denominator is zero there, so the function is undefined. The graph blows up to ยฑโ near x = 3.
Question
Common trap: a student confuses the asymptote y = 0 with an x-intercept. What is the difference?
Answer
x-intercept: the curve actually touches or crosses y = 0. Asymptote y = 0: the curve approaches y = 0 but never reaches it.
Question
f(x) = 5/(2x + 4). Find the vertical asymptote.
Answer
Set denominator = 0: 2x + 4 = 0 โ x = โ2. Vertical asymptote at x = โ2.
Question
What does "end behaviour" mean for a function?
Answer
How f(x) behaves as x โ โ or x โ โโ โ whether it grows, falls, or approaches a limiting value (asymptote).
Question
f(x) = 2 ยท 0.5หฃ. Describe the end behaviour as x โ โ.
Answer
As x โ โ, 0.5หฃ โ 0, so f(x) โ 0. The graph approaches the asymptote y = 0 from above and decreases toward it.
Question
A function increases without bound as x โ โ. How do you express this?
Answer
f(x) โ โ as x โ โ. There is no horizontal asymptote โ the function grows forever.
Question
IB asks "Describe the behaviour of the function for large values of x." What should your answer include?
Answer
State whether f increases, decreases, or approaches a fixed value. If it approaches a value, give the equation of the asymptote. Use context language if relevant.
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