Zeros of a Function

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Zeros of a Function: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Zeros of a function are the inputs that make the output exactly 0; on a graph they sit where the curve crosses or touches the x-axis.

Algebra 1Functions and GraphsA-APR.B.3

What is Zeros of a Function?

Ask which inputs make a rule return nothing at all, and you are hunting the zeros of a function. They solve \(f(x) = 0\), go by the name roots too, and each real one gives an x-intercept \((a, 0)\).

For a quadratic, factoring and the zero product property usually find them: if \(f(x) = (x + 1)(x – 3)\), the zeros are \(-1\) and 3. A function can have two, one or none. In context a zero often marks a break-even point or the moment something hits the ground.

How to spot it

  • A question asks where a curve reaches the horizontal axis, or for the roots of an equation.
  • A story asks when a quantity falls to zero, such as profit or height.
  • A function is already factored and the instructions ask for the zeros of a function.
  • A sign change in a table of values signals a zero between two inputs.

The method

  1. Write \(f(x) = 0\) and put the rule on the left.
  2. Factor the expression, or use the quadratic formula when it will not factor.
  3. Set each factor to zero and solve for x.
  4. Check each value by substitution to confirm that the output is 0.
(−1, 0)(3, 0)vertex (1, −4)f(x) = x² − 2x − 3= (x + 1)(x − 3)
f(x) = (x + 1)(x − 3) equals 0 at x = −1 and x = 3.

Zeros of a Function: solved example

Problem

A bakery’s profit in dollars is \(P(x) = -2x^2 + 28x – 48\), where x is dozens of cookies sold. Find the break-even points.

Solution

Break-even means zero profit, so solve \(-2x^2 + 28x – 48 = 0\). Divide by \(-2\): \(x^2 – 14x + 24 = 0\), which factors as \((x – 2)(x – 12) = 0\).

The zeros are \(x = 2\) and \(x = 12\). The bakery breaks even at 2 dozen and 12 dozen cookies, and earns a profit between those amounts.

Try it

Question 1

Find the zero of \(f(x) = 3x – 12\).

Show the answer

\(3x – 12 = 0\) gives \(x = 4\).

Question 2

Does \(f(x) = x^2 + 4\) have real zeros?

Show the answer

No. \(x^2 = -4\) has no real solution, so the graph never meets the x-axis.

Zeros of a Function vs. X-Intercept

One of the zeros of a function is a number, the input x. The intercept is the point \((x, 0)\) on the graph. For real zeros they carry the same information, and the answer’s format depends on the question: a value of x for a zero, a pair of coordinates for an intercept.

X-Intercept

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