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The zeros or roots of a function tell us where a graph intersects the x-axis. Since we’re talking about intersection with the x-axis, you know that y = 0. That means that you can find the zeros by solving the equation f(x) = 0.

Rule

Zeros

You find the zeros of a function by solving the equation

f(x) = 0.

The sign chart of f(x) tells you when the graph of f is above or below the x-axis, and where f(x) intersects the x-axis.

Graph showing f(x) and its zeros

Example 1

Find the zeros of f(x) = x2 + 5x + 6

You find the zeros by solving the equation f(x) = x2 + 5x + 6 = 0:

x = − 5 ±25 − 4 ⋅ 1 ⋅ 6 2 = − 5 ±25 − 24 2 = − 5 ± 1 2

This gives

x1 = − 5 − 1 2 = −3, x2 = − 5 + 1 2 = −2.

So the graph meets the x-axis at the points i (−3, 0) and (−2, 0).

Example 2

Find the zeros of g(x) = 2 sin ⁡ (2x) + 1 in the interval x ∈[0,π)

Again, you can find the zeros by solving the equation g(x) = 0:

2 sin ⁡ (2x) + 1 = 0 sin ⁡ (2x) = −1 2

The basic equation has the solutions

2x = −π 6 + 2πn, 2x = π −(−π 6 ) + 2πn.

Solving these for x, you get

x = −π 12 + πn, x = 7π 12 + πn.

Since the interval is x ∈[0,π), you get that x ∈{7π 12, 11π 12 }. The graph meets the x-axis at the points (7π 12, 0) and (11π 12 , 0).

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How to Calculate and Interpret Intersection Points

How to Calculate Zeros or Roots of a Function

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