The features exams ask for: "State the key features" just means: tell the story of the curve.
Where does it cross the axes?
Where are its high and low points?
What does it do at the far ends?
Answer those few questions and you've got it — the checklist below is the full menu.
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.
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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.
The curve from the worked example: it falls to a lowest point, then rises. Read the minimum's coordinates straight off — that turning point is the minimum.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
Local vs global: A local max/min is the highest/lowest in its immediate neighbourhood — like the top of one hill, even if a taller hill stands further along. A global max/min is the highest/lowest over the whole graph. A cubic has both a local max and a local min, but neither is global (the curve runs off to ±∞).
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
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.
Increasing where the curve goes uphill (left to right), decreasing where it goes downhill — a turning point is where it switches.
Interactive diagram
Explore the labelled diagram, charts and maps for this topic in full study mode.
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.
Plug in for values; graph for roots: For an unfamiliar function on Paper 2: to find f(a), just substitute the value — quick by hand, or type it straight into the GDC (see f(0) and f(16) below). The skill worth practising is the roots (where f(x) = 0): graph the function over its domain and use the zero / root tool to read off every x-intercept — the animation below walks through it.
IB-style question — values and roots
Consider f(x) = 6√x − x − 5, for 0 ≤ x ≤ 25. (a) Find (i) f(0); (ii) f(16). (b) Find the two roots of f(x) = 0.
Step by step
- (a)(i) Substitute x = 0.
- (a)(ii) Substitute x = 16, using √16 = 4.
- (b) Graph f over 0 ≤ x ≤ 25 and use the GDC's zero tool at each x-intercept.
Final answer
(a)(i) −5 (ii) 3. (b) x = 1 and x = 25.
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Set the window to the domain: Restrict the GDC window to the stated domain (here 0 ≤ x ≤ 25) before hunting for roots — a root outside the domain doesn't count, and the right window makes both x-intercepts easy to spot.