Graph Intercepts

X-intercepts and y-intercepts are where a graph meets the axes. Set y = 0 for x-intercepts, x = 0 for the y-intercept — worked through f(x) = x² + 1 (no real x-intercepts) and f(x) = x² − 1 (two of them).

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Intercepts are the points where a function’s graph meets the axes. Finding them turns out to need only one idea, applied twice.

The Idea On the axis, one coordinate is zero
y positive y negative
The sine curve crosses the x-axis at several points. At every one of them, — above the axis is positive, below it is negative. The same logic runs the other way: on the y-axis, everywhere, because there is no horizontal movement at all.
The Rule Set the other variable to zero

and represent the same output, so a function may be written or — either way, the rule is the same:

  • x-intercepts: set and solve for .
  • y-intercept: set and solve for .
Example 1
(0, 1)
The curve never dips down to the x-axis — it has no x-intercepts at all.
x-intercepts:
No real number squares to , so there are none.
y-intercept:

No x-intercepts, and a y-intercept at .

Example 2
(−1, 0) (1, 0) (0, −1)
This time the curve dips below zero before rising back up, so it crosses the x-axis twice.
x-intercepts:
y-intercept:

x-intercepts at and ; y-intercept at .

Summary
  1. On the x-axis, ; on the y-axis, .
  2. For x-intercepts, set and solve for .
  3. For the y-intercept, set and solve for .
  4. has no x-intercepts; has two, at .
285. Domain and Range287. Function Symmetry
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