The big idea: y-intercept: where the graph crosses the y-axis → this happens when x = 0.__LINEBREAK___x-intercept(s): where the graph crosses the x-axis → this happens when y = 0.__LINEBREAK__X-intercepts are also called roots or zeros of the function.
[Diagram: math-intercepts-explorer] - Available in full study mode
- y-intercept
- The point where the graph crosses the y-axis. Found by setting x = 0.
- x-intercept / root / zero
- A point where the graph crosses the x-axis. Found by setting y = 0.
| Intercept | x-value | y-value | How to find |
|---|---|---|---|
| y-intercept | 0 (always) | f(0) — calculate | Substitute x = 0 |
| x-intercept(s) | Solve f(x) = 0 | 0 (always) | Set function = 0 and solve |
The big idea: The y-intercept is found by substituting x = 0 into the function.__LINEBREAK__For any function in y = mx + c or y = ax² + bx + c form, the y-intercept is simply c — read it directly, no calculation needed.
y-intercept of a quadratic
Find the y-intercept of f(x) = 3x² − 2x + 5.
Step by step
- Substitute x = 0.
Final answer
y-intercept: (0, 5). Quick check: the constant term 5 matches — always read the constant term for y-intercept.
y-intercept of an exponential
Find the y-intercept of g(x) = 4 · 2ˣ − 1.
Step by step
- Substitute x = 0.
Final answer
y-intercept: (0, 3). Remember: any base raised to the power 0 equals 1.
Give intercepts as coordinates: IB expects you to write intercepts as coordinates.__LINEBREAK___✅ Correct: y-intercept is (0, 5)__LINEBREAK___❌ Incomplete: y-intercept is 5 (missing the x-coordinate)__LINEBREAK__The coordinate is (0, 5) — the x-value is always 0 at the y-intercept.
Quick read for standard form: If f(x) = ax² + bx + c → y-intercept is (0, c). Read c directly.__LINEBREAK__If f(x) = mx + c → y-intercept is (0, c). Read c directly.__LINEBREAK__No calculation needed for standard polynomial form.
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The big idea: To find x-intercepts: set f(x) = 0 and solve.__LINEBREAK__For linear functions: one solution. For quadratic functions: 0, 1, or 2 solutions. For higher-degree: can have more.__LINEBREAK__On Paper 2, you can use the GDC — find zeros using CALC > ZERO.
x-intercepts of a quadratic by factoring
Find the x-intercepts of f(x) = x² − 5x + 6.
Step by step
- Set f(x) = 0.
- Factorise.
- Solve each factor.
Final answer
x-intercepts: (2, 0) and (3, 0).
x-intercept of a linear function
Find the x-intercept of g(x) = 2x − 8.
Step by step
- Set g(x) = 0.
- Solve for x.
Final answer
x-intercept: (4, 0).
[Diagram: math-intercepts-explorer] - Available in full study mode
How many x-intercepts can there be?: Linear (y = mx + c): exactly 1, unless m = 0 (horizontal line)__LINEBREAK___Quadratic (y = ax² + bx + c): - Discriminant b² − 4ac > 0 → 2 intercepts - Discriminant = 0 → 1 intercept (tangent to x-axis) - Discriminant < 0 → 0 intercepts (never crosses)__LINEBREAK___Exponential (y = abˣ + c): usually 0 or 1, depending on c.
GDC method (Paper 2): 1. Enter the function into your GDC. 2. Set a window that shows where the curve crosses the x-axis. 3. Use CALC > ZERO (or equivalent), set left/right bounds around the root. 4. Record the x-value displayed — write it to 3 significant figures unless told otherwise. 5. If there are two roots, repeat for each one.
The big idea: In real-world models, intercepts have physical meaning.__LINEBREAK__The y-intercept = the value of the quantity when x = 0 (the starting value, the initial amount, the value at time zero).__LINEBREAK__The x-intercept = the value of x when the quantity is zero (when something runs out, ends, or hits ground level).
Interpreting intercepts in a revenue model
A company's monthly revenue is modelled by R(p) = −2p² + 100p, where p is the price in euros. Find and interpret both intercepts.
Step by step
- y-intercept (p = 0): R(0) = 0. Revenue is zero when price is zero.
- x-intercepts (R = 0): factorise.
- Solve: p = 0 or p = 50.
Final answer
At p = 0 (giving the product away free) and at p = 50 (price too high, no buyers) revenue is zero. Between these, revenue is positive.
| Intercept | What it tells you in context |
|---|---|
| y-intercept | The initial value at x = 0 (start, time zero, price zero, etc.) |
| x-intercept | When the quantity equals zero (revenue = 0, height = 0, amount runs out) |
| Both intercepts | Together they frame the "active" domain of the model |
Write a full context sentence: IB awards a separate mark for interpreting an intercept in context.__LINEBREAK___✅ Correct: "The x-intercept at p = 50 means the revenue is zero when the price is €50, because no customers buy at that price."__LINEBREAK___❌ Incomplete: "x-intercept is 50" — no interpretation of what 50 means in the situation.