Increasing and decreasing intervals
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Flip to reveal answersDefine an increasing function on an interval.
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All 16 Flashcards — Increasing and decreasing intervals
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Question
Define an increasing function on an interval.
Answer
f is increasing on an interval if the output rises as you move left to right: whenever x₁ < x₂, we have f(x₁) < f(x₂). The graph goes upward.
Question
How can you tell a function is decreasing from its graph?
Answer
The graph moves downward as you read from left to right — outputs fall as inputs increase.
Question
At a local maximum, is the function increasing or decreasing immediately before it?
Answer
Increasing — the function rises up to the maximum, then begins decreasing after it.
Question
What notation does IB accept for stating intervals?
Answer
Inequalities (e.g. 1 < x < 4) and interval notation (e.g. (1, 4)) are both accepted. Write whichever matches the question's phrasing.
Question
A graph rises from x = −2 to x = 1, then falls. On what interval is f increasing?
Answer
f is increasing on −2 < x < 1 (or [−2, 1]).
Question
A function has a maximum at x = 2 and minimum at x = 5. State all increasing and decreasing intervals.
Answer
Increasing: x < 2 and x > 5. Decreasing: 2 < x < 5.
Question
IB asks "State the interval on which f is decreasing." What format is required?
Answer
An inequality or interval notation including both endpoints. E.g. 2 ≤ x ≤ 5 or [2, 5]. The interval must refer to x-values (inputs), not y-values.
Question
f(x) = x². On what interval is f decreasing?
Answer
For x < 0. The parabola falls from left toward x = 0, then rises for x > 0. The minimum is at (0, 0).
Question
A student writes "f is increasing at x = 3." What is wrong?
Answer
"Increasing at a point" is meaningless. Increasing is a property of an interval, not a single point. Write "f is increasing for x > 3" or "f is increasing on (1, 3)".
Question
IB asks for the "interval on which f is increasing." A student writes "f(x) increases from 4 to 9." What is wrong?
Answer
The answer should be an interval of x-values, not y-values. Correct: e.g. "1 < x < 3." The y-values (4 to 9) are outputs, not the interval.
Question
Should you include the endpoints of a turning point in an increasing interval? E.g. is the max at x = 2 included?
Answer
IB accepts both x < 2 and x ≤ 2 for the increasing interval up to a maximum. Either strict or inclusive inequalities are fine unless the question specifies.
Question
A linear function y = 3x − 1. Is it increasing, decreasing, or neither?
Answer
Increasing everywhere — gradient is 3 > 0, so the output always rises as x increases. No turning points.
Question
T(t) is increasing for 0 ≤ t ≤ 5 (hours). What does this mean in context?
Answer
The temperature rises during the first 5 hours.
Question
IB asks "Find the intervals during which the population is decreasing." What type of answer is needed?
Answer
An interval of t-values (the input variable), e.g. "3 < t < 8 hours." Not y-values. Use the same variable as the context.
Question
Profit increases from n = 0 to n = 200, then decreases. What is significant about n = 200?
Answer
n = 200 is where the profit function has its local maximum — the production level giving the greatest profit.
Question
IB asks "Describe the behaviour of f for large positive values of x." What kind of answer is needed?
Answer
State whether f is increasing or decreasing, and whether it approaches a fixed value (asymptote) or continues without bound. E.g. "f is decreasing and approaches y = 3."
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