Key takeaway: The zeros of a function are the input values x for which f(x) equals 0, and on the graph they are the points where the curve meets the x-axis.
Grade 9 – Algebra 1A-APR.B.3Functions and Graphs
Definition of Zeros of a Function
The zeros of a function are the inputs that make its output exactly zero. They are also called roots when you talk about an equation, and x-intercepts when you talk about the graph. The three names describe the same numbers from different angles.
To find them, replace \(f(x)\) with 0 and solve. For a quadratic that factors, factor first and use the zero product property. For other quadratics, use the quadratic formula or completing the square. A function can have no real zeros, one, or several, depending on its shape. Zeros mark the break-even point in a business model, or the moment a falling object hits the ground.
Formula and notation
A number \(r\) is a zero of \(f\) when
\[f(r) = 0\]
The point on the graph is \((r, 0)\). If \(f(x) = (x – r)(x – s)\), the zeros are \(r\) and \(s\), and the factor \((x – r)\) corresponds to the zero \(r\).
Key facts
- Zeros of a function, roots of an equation and x-intercepts of a graph are the same values.
- The zeros of a function of degree 2 number at most two.
- The factor (x − r) gives the zero r with the opposite sign.
- Substituting a zero into the function must return exactly 0.
Zeros of a Function example
Find the zeros of \(f(x) = 2x^{2} – x – 10\).
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Set the output to zero: \(2x^{2} – x – 10 = 0\).
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Factor: \((2x – 5)(x + 2) = 0\). Check: \(2x^{2} + 4x – 5x – 10 = 2x^{2} – x – 10\).
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Split: \(2x – 5 = 0\) gives \(x = \frac{5}{2}\), and \(x + 2 = 0\) gives \(x = -2\).
The zeros are \(x = -2\) and \(x = \frac{5}{2}\).
Common mistake
Watch out
Cora sees the factor \((x + 2)\) and says that \(x = 2\) is a zero of the function.
Why it is wrong
A zero makes its factor equal 0: \(x + 2 = 0\) gives \(x = -2\). Substituting her value shows the error: \(f(2) = 8 – 2 – 10 = -4\), not 0, while \(f(-2) = 8 + 2 – 10 = 0\). The sign flips when you solve the factor.
Related terms and lessons
- Algebra 1 · Solving Quadratic EquationsLessonWorksheetQuiz
- Algebra 2 · Polynomial Functions and Their GraphsLessonWorksheetQuiz
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