What does a function's zero tell me, and how can I find it?
Quick answer
A function’s zero is an input value that makes its output 0: if x is a zero, then f(x)=0. On the graph, that value is the x-coordinate where the curve meets the horizontal axis. For a factored formula, set each factor equal to zero and solve. For example, (x−2)(x+1)=0 gives zeros 2 and −1, shown at (2,0) and (−1,0). [S1]
The question, the details, the next step.
What does it mean when a function has a zero? Is the zero the point on the graph, or something else?
A zero is an input, usually called x, that makes the function’s output equal to 0. In symbols, a zero is a value of x for which f(x)=0. [S1]
The zero itself is the x-value, not the whole point. On a graph, it corresponds to the point where the curve meets the horizontal axis: the point’s coordinates are (zero, 0). For instance, if x=3 is a zero, the corresponding graph point is (3,0).
A useful way to keep the two views straight: the formula tells you which input makes the output zero; the graph shows that input as a horizontal-axis intersection. A function may have more than one zero.
Can you show the algebra steps for finding them from a factored formula?
Take the example f(x)=(x−2)(x+1). To find its zeros, set the output equal to zero:
- (x−2)(x+1)=0
- Set each factor equal to zero: x−2=0 or x+1=0.
- Solve: x=2 or x=−1.
So the zeros are 2 and −1. This works because a product is zero when at least one of its factors is zero. The example follows the factored-form method: set the function equal to zero, then solve each factor separately. [S1]
Check by substitution: at x=2, the first factor is 0; at x=−1, the second factor is 0. Either way, the product—and therefore f(x)—is 0. That check can catch a sign error.
How do those answers show up on the graph, and what if I only have a graph?
For this same function, the zeros −1 and 2 correspond to the graph’s x-intercepts (−1,0) and (2,0). [S1] The zero is each point’s x-coordinate; the y-coordinate is 0 because the point lies on the horizontal axis.
If you only have a graph, look for where its curve meets the horizontal axis, then read the x-coordinate at each intersection. For example, if a plotted curve meets the axis at (−1,0) and (2,0), its zeros are −1 and 2. This is the graphical counterpart of solving f(x)=0.
If the graph is a rough sketch without readable scale marks, it may not give you an exact numerical zero.
Does the graph have to cross the horizontal axis for an x-value to be a zero?
No. The key is whether the graph meets the horizontal axis, not whether it crosses from one side to the other. At any point on that axis, the output is 0, so the point’s x-coordinate is a zero. [S1]
For example, if a curve touches the axis at (2,0) and turns back without crossing, x=2 is still a zero. When reading a graph, include every point where the curve meets the axis, and report its x-coordinate. If the point is between marked grid values, describe the value as an estimate rather than an exact number.
A practical checklist
Set the function’s output equal to 0.
For a factored formula, solve each factor equal to 0.
Substitute each proposed zero to check that the output is 0.
On a graph, locate horizontal-axis intersections and read their x-coordinates.
State zeros as x-values and intercepts as coordinate pairs (x,0).
A few more useful answers
Is a zero the same thing as an x-intercept?
They refer to the same location in two different ways: the zero is the x-value, while the x-intercept is the point (x,0) on the graph. [S1]
How many zeros can a function have?
A function can have one or more zeros, or none. On a graph, look for points where it meets the horizontal axis; from a formula, solve f(x)=0.
What should I do if the graph’s intersection is between grid marks?
Read the x-coordinate as an estimate and label it as approximate. A rough graph may not support an exact zero; use an exact formula or clearer scale if one is available.
Sources
References used for this guide.
- [S1] OpenStax College Algebra: Graphs of Polynomial Functions openstax.org · Checked