Determine the zeros of a quadratic relation from a graph

Zeros of a quadratic relation are the real $x$ -values for which $y=0$ , represented by the points where its parabola intersects or touches the $x$ -axis. The interpretation includes recognizing two distinct zeros, one repeated zero at the vertex, or no real zeros, and reading exact or approximate values from an appropriately scaled graph without confusing zeros with the $y$ -intercept; complex zeros and more advanced algebraic treatments are not included.

Detailed Explanation: Determine the zeros of a quadratic relation from a graph

A zero is an $x$ -value where the quadratic has $y=0$ . On a graph, zeros are the points where the parabola intersects or touches the $x$ -axis.

Example

Suppose a graph of a parabola shows that it crosses the $x$ -axis at the points

(-2,0) \quad \text{and} \quad (3,0).

Step 1: Find where the parabola meets the $x$ -axis.
The $x$ -axis is the line $y=0$ . The graph meets it at two points.

Step 2: Read the $x$ -coordinates.
The $x$ -coordinates of the points are $-2$ and $3$ .

Step 3: State the zeros.

\boxed{x=-2 \text{ and } x=3}

The zeros are the $x$ -values, not the full points.

Remember:

If the parabola crosses the $x$ -axis twice, it has two distinct zeros.
If it only touches the $x$ -axis at its vertex, it has one repeated zero.
If it never reaches the $x$ -axis, it has no real zeros.

Do not use the point where the parabola crosses the $y$ -axis; that point is the $y$ -intercept, not a zero unless it also lies on the $x$ -axis.

Learn by doing: Determine the zeros of a quadratic relation from a graph

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Quadratic Discriminants - Graph to Root Example

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