Key Idea: A graph is a visual summary of a function's behaviour. Topic 2.3 covers reading and drawing graphs — finding where they cross axes, where they intersect each other, and using the GDC effectively to get this information quickly. The ability to set an appropriate viewing window and read off values accurately is central to almost every Paper 2 question.
✅ Key graph features
- x-intercept (root)
- Where the graph crosses the x-axis. Found by setting f(x) = 0 and solving, or using GDC 'zero' function. Example: f(x) = x² − 5x + 6 has x-intercepts at x = 2 and x = 3.
- y-intercept
- Where the graph crosses the y-axis. Found by substituting x = 0. Example: f(0) = 0 − 0 + 6 = 6, so y-intercept is (0, 6).
- Intersection of two graphs
- The points where two functions have the same output. Found graphically (GDC intersect function) or algebraically (set equal and solve). The intersection (x, y) satisfies both equations simultaneously.
📊 GDC graphing workflow
- Step 1: Set the window
- Adjust X-min, X-max, Y-min, Y-max. If the graph does not show the key features, zoom out or use ZoomFit.
- Step 2: Find x-intercepts
- Use 2nd → Calc → Zero (TI) or G-Solve → Root (Casio). Move the cursor to each root.
- Step 3: Find intersections
- Graph both functions. Use 2nd → Calc → Intersect (TI) or G-Solve → Intersect (Casio). The GDC gives both x and y coordinates.
Example: Find x-intercepts of f(x) = 2x² − 3x − 5: GDC finds roots at x = −1 and x = 2.5 (or solve: (2x−5)(x+1)=0) Find where f(x) = x² and g(x) = 2x + 3 intersect: Graph both. GDC intersect → x = −1 (y = 1) and x = 3 (y = 9).
When reading intercepts from a graph: always double-check by substituting back. A GDC that shows a root at x = 2.00 should be confirmed as exact (check f(2) = 0) or written as approximate. Do not confuse x-intercept (y = 0) with y-intercept (x = 0). Both are visible on the graph but come from different substitutions.
Paper 2 (GDC allowed): Always sketch the graph in your answer, even roughly. Mark the intercepts and intersections you found. This earns method marks and helps you avoid misreading the GDC output. Paper 1 (GDC allowed): You will be given the graph and asked to read off coordinates, or given the equation and asked to find intercepts algebraically.
IB-style question [7 marks]
A drone is launched from the ground and its height, d metres, t minutes after launch is modelled by d(t) = −t² + 6t, for 0 ≤ t ≤ 6. (a) Find the y-intercept of the graph of d and state what it represents. (b) Find the x-intercepts of the graph of d. (c) Find the maximum height of the drone and the time at which it occurs. (d) Sketch the graph of d for 0 ≤ t ≤ 6, labelling the intercepts and the maximum point.
Step by step:
(a) The y-intercept is d(0) — substitute t = 0.
So the graph passes through (0, 0): the drone starts on the ground.
(b) The x-intercepts are where d(t) = 0. Set the model to zero and factorise.
Each factor gives a solution.
(c) The maximum is halfway between the two x-intercepts.
Substitute t = 3 to get the maximum height.
(d) Draw a smooth ∩-shaped curve through (0, 0), the vertex (3, 9) and (6, 0), labelling each with its coordinates.
(a) (0, 0) — the drone starts on the ground. (b) t = 0 and t = 6. (c) Maximum height 9 m at t = 3 minutes. (d) ∩-shaped curve through (0, 0), (3, 9) and (6, 0).