Explaining zeros of a function for students
Imagine a game score that can go up and down and is shown as a line on a graph. The zeros of a function are the points where that line hits the ground level, which is the x-axis. At those x-values, the function’s value is exactly 0. They are also called roots or solutions of the related equation.
Worked examples with zeros of a function
Example 1 (basic, linear function)
Find the zeros of the function f(x) = 2x − 6.
To find zeros, set the function equal to 0 and solve for x:
2x − 6 = 0
Add 6 to both sides: 2x = 6.
Divide both sides by 2: x = 3.
So the function has one zero at x = 3. On the graph, the line crosses the x-axis at the point (3, 0).
Example 2 (advanced, quadratic function and factoring)
Find the zeros of the function g(x) = x² − 5x + 6.
First, factor the quadratic expression x² − 5x + 6. Look for two numbers that multiply to 6 and add to −5. Those numbers are −2 and −3, so:
x² − 5x + 6 = (x − 2)(x − 3).
To find zeros, set g(x) = 0:
(x − 2)(x − 3) = 0.
Use the zero product property, which says that if a product is 0, at least one factor must be 0. So either x − 2 = 0 or x − 3 = 0.
If x − 2 = 0, then x = 2. If x − 3 = 0, then x = 3.
The zeros of the function are x = 2 and x = 3. On the graph of g(x), the parabola crosses the x-axis at (2, 0) and (3, 0). In equation form, these are also the solutions of x² − 5x + 6 = 0.
How zeros of a function connect to other ideas
In algebra, zeros of a function are closely tied to factors and solutions of equations. For a polynomial written in factored form, like f(x) = (x − 1)(x + 4), each linear factor (x − a) gives a zero at x = a, and these same values are the solutions of the equation f(x) = 0.
For quadratic functions, the zeros match the x-intercepts of the graph and the solutions of the related quadratic equation. Students may find these zeros by factoring, by using the quadratic formula, or by using technology to approximate them when they are not integers.
In higher grades, students extend this idea to more complicated polynomial and rational functions. They learn that a polynomial of degree n has exactly n complex zeros (counting repeats), and that knowing the zeros helps sketch the graph and understand key features like where the graph crosses or touches the x-axis.
Common mistakes with zeros of a function
- Mistake: Thinking the zeros are the y-values where the graph crosses the x-axis, for example saying the zero is 0 instead of x = 3 when the point is (3, 0). Correction: Zeros are x-values (inputs), not y-values; they are the x-coordinates where the function’s output is 0.
- Mistake: Forgetting to set the function equal to 0 before solving, for example solving 2x − 6 = 10 instead of 2x − 6 = 0 when asked for zeros. Correction: To find zeros, always solve the equation f(x) = 0, because zeros are where the output equals 0.
- Mistake: Stopping after factoring a quadratic and not solving each factor, for example factoring x² − 5x + 6 as (x − 2)(x − 3) and then saying the zeros are (x − 2) and (x − 3). Correction: After factoring, set each factor equal to 0 and solve to get the actual zero values, x = 2 and x = 3 in this case.
Where zeros of a function show up in everyday life
Zeros of a function show up any time a quantity hits a balance point of 0. A height function for a ball thrown into the air has zeros when the ball leaves and returns to the ground. A profit function has zeros at the break-even points where money made and money spent are equal. On a graph, those moments are exactly where the curve meets the x-axis.