Big idea: A root is where the graph hits the x-axis.
At that point, the y-value is 0.
IB often uses the words root, zero, solution, and x-intercept almost interchangeably.
Quick example
A graph crosses the x-axis at x = 3.
What is the root?
Step by step
- The root is the x-value where the graph meets the x-axis.
- The graph crosses at x = 3.
Final answer
Root = 3
Animated graph
Watch the graph build step by step in study mode.
| Question asks for... | You give... |
|---|---|
| Root | x = 3 |
| x-intercept | (3, 0) |
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Using 2:zero on your GDC
What 2:zero does: The zero tool finds an x-value where y = 0.
You set Left Bound, Right Bound, then Guess.
Use the zero tool one root at a time.
If the graph has two crossings, run the tool twice.
Quick example
Use technology to solve x² − 3 = 0 and give the positive root.
Step by step
- Enter y = x² − 3 and graph it.
- Press 2nd TRACE 2:zero.
- Set a left bound and right bound around the positive crossing.
- Press ENTER for guess and read the x-value.
Final answer
Positive root: x ≈ 1.73
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
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Technology finds roots fast: A root is an x-value where y = 0, where the graph crosses or touches the x-axis.
Your GDC tool for this is 2nd -> TRACE -> 2:zero.
What is a polynomial graph?: A polynomial graph is built from powers of x.
Different polynomial types create different graph shapes and different numbers of roots.
| Graph type | Example | Graph behaviour |
|---|---|---|
| Linear | y = 2x + 1 | Usually crosses the x-axis once -> 1 root |
| Quadratic | y = x2 - 4 | Can cross twice, touch once, or miss the x-axis completely |
| Cubic | y = x3 - x | Can have up to 3 real roots because it can cross the x-axis multiple times |
| Polynomial | Any graph made from powers of x | More turning points can create more x-axis crossings and more roots |
What the zero tool gives you: Find a root -> where the graph crosses the x-axis. Get an estimate -> decimal x-value. Find more roots -> run 2:zero again, one crossing at a time.
Worked example
Use your GDC to find both roots of x2 - 5x + 4 = 0.
Step by step
- Enter the function:
- Press GRAPH. The curve crosses the x-axis twice, so there are two real roots.
- Use 2nd -> TRACE -> 2:zero near the first crossing to get the first root.
- Run 2:zero again near the second crossing to get the second root.
Final answer
x = 1 and x = 4
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
IB exam habits: The zero tool finds one root at a time. If there are multiple crossings, run zero again for each one. Round exactly as the question asks. Report the root as the x-value, not a full coordinate, unless asked.
Calculator roots are often decimals: IB usually asks you to round approximate roots to a certain number of decimal places or significant figures.
Worked example
The calculator gives x = −2.418736 as a root.
Report it to 3 significant figures.
Step by step
- Keep the first 3 significant digits.
- Look at the next digit to decide whether to round up.
Final answer
x ≈ −2.42
Use approximation notation: If the root is rounded, use ≈ instead of =.