The features exams ask for: "State the key features" almost always means some of: intercepts (where it crosses the axes), maximum / minimum points, asymptotes, where it is increasing / decreasing, and its behaviour as x → ±∞.
Features to read off
- Intercepts — crosses the axes (y = 0 or x = 0).
- Max / min — the turning points.
- Asymptotes — lines it approaches.
...and the behaviour
- Increasing / decreasing — going up or down.
- End behaviour — what happens as x → ±∞.
- Symmetry — about a line or the origin.
Answer exactly what's asked: "Maximum value" wants a y-coordinate; "maximum point" wants coordinates (x, y); "where the maximum occurs" wants the x-coordinate. Read the wording.
Zero = root = x-intercept: The x-intercepts are where y = 0 — also called the zeros or roots. The y-intercept is where x = 0. Three words, one idea for the x-axis crossings.
IB-style question — intercepts of a function
Find the intercepts of f(x) = (x − 2)(x + 4).
Step by step
- Zeros: set f(x) = 0 (each factor).
- y-intercept: set x = 0.
Final answer
Zeros at x = 2 and x = −4; y-intercept (0, −8).
Watch the language: If a question says "find the zeros" or "solve f(x) = 0" or "find where the graph cuts the x-axis", it's all the same task.
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Turning points: where the curve turns: A maximum or minimum is a turning point — where the curve stops rising and starts falling, or vice versa. The value is its y-coordinate; the point is the full (x, y).
IB-style question — read a min off vertex form
State the minimum point and minimum value of f(x) = (x − 3)² − 4.
Step by step
- Vertex form a(x − h)² + k has its turning point at (h, k).
- a = 1 > 0, so it's a minimum.
Final answer
Minimum point (3, −4); minimum value −4.
Local vs global: A local max/min is the highest/lowest in its immediate neighbourhood; a global one is the highest/lowest over the whole graph. A cubic can have a local max and a local min.
Lines the curve heads toward: A vertical asymptote is where the curve shoots to ±∞ — where a denominator = 0. A horizontal asymptote is the value y approaches as x → ±∞ (the curve levels off).
IB-style question — asymptotes of a rational graph
State the asymptotes of f(x) = 2 + 1/(x − 5).
Step by step
- Vertical: denominator zero.
- Horizontal: as x → ±∞ the fraction → 0, leaving the +2.
Final answer
Vertical asymptote x = 5; horizontal asymptote y = 2.
End behaviour in words: "As x → ∞, y → 2" describes the curve flattening toward its horizontal asymptote. Exams often want this stated, not just the line drawn.
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Up, down, and mirror lines: A function is increasing where the graph goes up left-to-right, and decreasing where it goes down — give the x-intervals. Symmetry: a parabola is symmetric about its vertical axis; an even function about the y-axis.
IB-style question — where is it increasing?
For the parabola f(x) = x² − 4, state where f is increasing and where it is decreasing.
Step by step
- The vertex (turning point) is at x = 0.
- Left of the vertex the curve falls; right of it the curve rises.
Final answer
Decreasing for x < 0, increasing for x > 0.
Turning points split the intervals: Increasing/decreasing change at the turning points. Find the max/min x-values first, then say what happens on each side.