To solve a quadratic equation by graphing, graph the related function . The real solutions are represented by the -intercepts of the graph. A quadratic function may have zero, one, or two -intercepts. For example:
- For the equation , the graph of its related function, , has two -intercepts, and . So, the equation has two real solutions: and .
- For the equation , the graph of its related function, , has one -intercept at . So, the equation has one real solution at .
- For the equation , the graph of the related function, , has no -intercepts. So, the equation has no real solutions.
|Solutions or Roots||Zeros||-Intercepts|
|An equation has solutions, which are also called roots. A solution is a value of the variable that makes the equation true.||A function has zeros. A zero is an input value that makes the value of the function zero.||A graph has -intercepts. An -intercept is an -coordinate of a point where the graph touches or crosses the -axis.|
Use the values of , , and to determine the vertex and axis of symmetry.
Another point that is easy to locate is 1 unit right of the vertex. From the vertex, move 1 unit right and units up (if is positive) or down (if is negative). Since , move 1 unit right and 1 unit up from the vertex. The graph passes through .
Use symmetry to identify points reflected across the axis of symmetry. The axis of symmetry is , so the mirror images of the points and with respect to axis of symmetry are also points on the graph. Therefore, the graph passes through and .
The -intercepts represent the solutions of the equation. Use the graph to identify the -intercepts.The graph appears to pass through the points and , so the -intercepts appear to be –5 and –1.
Verify the solutions by substituting the values of the -intercepts in the original quadratic equation. Using the point :