The big idea: y-intercept: where the graph crosses the y-axis → this happens when x = 0.
x-intercept(s): where the graph crosses the x-axis → this happens when y = 0.
X-intercepts are also called roots or zeros of the function.
Drag the function curve. Watch the x-intercepts and y-intercept update automatically. Notice that some functions have no x-intercept.
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- 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 |
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The big idea: The y-intercept is found by substituting x = 0 into the function.
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.
✅ Correct: y-intercept is (0, 5)
❌ Incomplete: y-intercept is 5 (missing the x-coordinate)
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.
If f(x) = mx + c → y-intercept is (0, c). Read c directly.
No calculation needed for standard polynomial form.
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The big idea: To find x-intercepts: set f(x) = 0 and solve.
For linear functions: one solution. For quadratic functions: 0, 1, or 2 solutions. For higher-degree: can have more.
You can use the GDC on either paper — 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).
Enter f(x) = x² − 5x + 6 and verify the two x-intercepts at (2, 0) and (3, 0). Then try a function with no x-intercepts.
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How many x-intercepts can there be?: Linear (y = mx + c): exactly 1 x-intercept (unless the line is perfectly horizontal — then 0 or infinitely many).
Quadratic (y = ax² + bx + c) — the U-shape can sit three ways: • Cuts the x-axis at 2 points — most common. • Sits just touching the x-axis at 1 point — the vertex is on the axis. • Sits entirely above or entirely below the x-axis — 0 intercepts, the curve never crosses.
A quick sketch (vertex + opening direction) tells you the count before any algebra.
GDC method: 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.
The y-intercept = the value of the quantity when x = 0 (the starting value, the initial amount, the value at time zero).
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.
Bank-balance context — when does the account run out?
Maya opens a savings account with $400 and withdraws $25 each week to cover lunches.
Her balance after t weeks is B(t) = 400 − 25t.
Find both intercepts and explain what each means.
Step by step
- y-intercept: set t = 0.
- x-intercept: set B(t) = 0 and solve.
Final answer
y-intercept (0, 400): Maya starts with $400 in the account. x-intercept (16, 0): the account hits $0 after 16 weeks of withdrawals — that is when the money runs out.
| 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.
✅ Correct: "The x-intercept at p = 50 means the revenue is zero when the price is €50, because no customers buy at that price."
❌ Incomplete: "x-intercept is 50" — no interpretation of what 50 means in the situation.