The big idea: A graph shows input x on the horizontal axis and output y on the vertical axis.
Each point (x, y) on the graph means y = f(x) for that x.
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
IB language: Read from x to y when asked for f(a). Find x = a on the horizontal axis, move up or down to the line, then read the y-value.
The big idea: Plot key points first, then draw smooth or straight connections based on function type.
Example: Sketching a linear function
Sketch y = 2x + 1 for x = −1, 0, 1, 2.
STEPS
- Build a value table.
- Mark y-intercept: the line crosses the y-axis at (0, 1).
- Use the gradient m = 2: go right 1, up 2, to reach (1, 3).
- Plot all four points and draw a straight line through them.
Final answer
A straight line through (−1, −1), (0, 1), (1, 3), (2, 5).
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
The same two-step method works for any linear function. Try the lines below:
[Diagram: math-sketch-from-m-and-c] - Available in full study mode
IB sketch rule: You only need two points to draw an exact straight line — but plotting a third point is a free check. If all three are collinear you have no arithmetic error.
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The big idea: To find f(a): move to x = a on the horizontal axis, go straight up to the curve, then read the y-value.__LINEBREAK__To find x when f(x) = k: move to y = k on the vertical axis, go across to the curve, then read the x-value(s).__LINEBREAK__These are the two graph-reading skills IB tests most often.
Reading f(a) from a graph
The graph of f is given. Using the graph, find f(3) and find x when f(x) = 0.
Step by step
- For f(3): start at x = 3 on the x-axis. Go vertically up to the curve.
- The y-value at the curve above x = 3 is 5.
- For f(x) = 0: start at y = 0 (the x-axis) and find where the curve touches it.
- The curve crosses y = 0 at x = −2 and x = 4.
Final answer
f(3) = 5. The function equals zero when x = −2 or x = 4.
IB tolerance on graph reading: When reading from a graph (Paper 1), IB usually accepts answers within ±0.2 of the exact value.__LINEBREAK__If the curve passes through exactly (3, 5), answers of 4.8 to 5.2 are accepted.__LINEBREAK__Use a ruler and read carefully.
[Diagram: math-function-grapher] - Available in full study mode
Reading x-values vs y-values: "Find f(3)" → gives a y-value (an output).__LINEBREAK___"Find x when f(x) = 5" → gives an x-value (an input).__LINEBREAK__These are opposite directions on the graph. Never confuse which axis you start from.
The big idea: IB uses the same five function families repeatedly. If you can identify the shape from a graph, you can state the function type instantly — without algebra.__LINEBREAK__Learn the characteristic shape of each family.
| Family | Typical shape | Key feature to spot |
|---|---|---|
| Linear y = mx + c | Straight line | No curve at all |
| Quadratic y = ax²+bx+c | U-shape (a>0) or ∩-shape (a<0) | One turning point, symmetric |
| Exponential y = abˣ | Rapid growth/decay curve | Horizontal asymptote, never crosses it |
| Power y = axⁿ | Curve through origin or near it | Depends on n: cubic has inflection point |
| Sinusoidal y = a sin(bx)+c | Wave, repeating equally | Regular peaks and troughs, periodic |
[Diagram: math-function-grapher] - Available in full study mode
Identifying the family from a graph
A graph shows a curve that starts high, decreases, and approaches but never crosses the x-axis. What function family is it most likely?
Step by step
- The curve approaches the x-axis without crossing it.
- This means y → 0 as x → ∞ — a horizontal asymptote at y = 0.
- A horizontal asymptote with exponential decay shape → exponential family.
Final answer
Exponential decay: y = abx where 0 < b < 1.
Exam recognition shortcut: In the exam, these four features quickly identify a family:__LINEBREAK___🔵 No curve → linear 🔵 One turning point, symmetric → quadratic 🔵 Asymptote, never crosses → exponential 🔵 Regular waves → sinusoidal__LINEBREAK__State the family first, then find parameters.