Understanding the X-Intercept of a Quadratic Function

Hand elevating graph line off the page
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The graph of a quadratic function is a parabola. A parabola can cross the x-axis once, twice, or never. These points of intersection are called x-intercepts. Does this concept sound familiar, yet strange? Your teacher may call these points by their nicknames.

Other Terms for x-intercepts

  • Zeros
  • Roots
  • Solutions
  • Solution set

Four Methods of Finding the x-intercepts

A Parabola with Two X-intercepts

Use your finger to trace the green parabola. Notice that your finger touches the x-axis at (-3,0) and (4,0).

Therefore, the x-intercepts are (-3,0) and (4,0)

Be careful:  the x-intercepts are not merely -3 and 4. The answer should be an ordered pair. Notice that the y-value of of these points is always 0.

A Parabola with One x-intercept

Parabola with one root
Krishnavedala/Wikimedia Commons/Creative Commons 3.0

Use your finger to trace the blue parabola. Notice that your finger touches the x-axis at (3,0).

Therefore, the x-intercept is (3,0).

Question: When a parabola has only one x-intercept, is the vertex always the x-intercept?

A Parabola Without x-intercepts

Parabola with no x intercept
Olin/Wikimedia Commons/Creative Commons 3.0

Use your finger to trace the blue parabola. Does your finger touch the x-axis? No. Therefore, this parabola has no x-intercepts.