The x‑intercept is where a graph crosses the x‑axis, which is the point where y=0y=0y=0.

Quick Scoop: Core Idea

  • On a graph : The x‑intercept is the point where the curve touches or crosses the horizontal axis (x‑axis), so its coordinates look like (a,0)(a,0)(a,0).
  • With an equation : You always find the x‑intercept(s) by setting y=0y=0y=0 and solving for xxx.

Think of it like this: the x‑axis is the line where the vertical value is zero, so to hit that line, your yyy must be 0.

1. From a line in slope‑intercept form y=mx+by=mx+by=mx+b

Steps:

  1. Set y=0y=0y=0.
  2. Solve the resulting equation for xxx.

Example: y=2x−4y=2x-4y=2x−4

Set y=0y=0y=0:
0=2x−40=2x-40=2x−4
Add 4: 4=2x4=2x4=2x
Divide by 2: x=2x=2x=2 So the x‑intercept is the point (2,0)(2,0)(2,0).

2. From standard form Ax+By+C=0Ax+By+C=0Ax+By+C=0

Steps:

  1. Set y=0y=0y=0.
  2. Solve Ax+C=0Ax+C=0Ax+C=0 for xxx.

Algebraically, Ax+C=0⇒x=−C/AAx+C=0\Rightarrow x=-C/AAx+C=0⇒x=−C/A.

So the x‑intercept is (−C/A,0)(-C/A,0)(−C/A,0).

Example: 3x+2y−6=03x+2y-6=03x+2y−6=0
Set y=0y=0y=0: 3x−6=03x-6=03x−6=0
Solve: 3x=6⇒x=23x=6\Rightarrow x=23x=6⇒x=2
X‑intercept: (2,0)(2,0)(2,0).

3. From a quadratic (parabola) y=ax2+bx+cy=ax^2+bx+cy=ax2+bx+c

Quadratics can have 0, 1, or 2 x‑intercepts.

Steps:

  1. Set y=0y=0y=0: 0=ax2+bx+c0=ax^2+bx+c0=ax2+bx+c.
  2. Solve the quadratic for xxx (factoring, completing the square, or quadratic formula).

Example: y=x2−3x+2y=x^2-3x+2y=x2−3x+2

Set y=0y=0y=0:
0=x2−3x+20=x^2-3x+20=x2−3x+2
Factor: (x−1)(x−2)=0(x-1)(x-2)=0(x−1)(x−2)=0
So x=1x=1x=1 or x=2x=2x=2. X‑intercepts: (1,0)(1,0)(1,0) and (2,0)(2,0)(2,0).

4. From just a graph (no equation)

Look at the picture of the graph and see where it crosses the x‑axis.

  • If it crosses once: one x‑intercept.
  • If it touches and bounces off: still one x‑intercept (a “double root”).
  • If it never touches the x‑axis: no x‑intercepts.

Read the xxx value at each crossing; that gives you the x‑intercept point(s) (x,0)(x,0)(x,0).

5. Tiny “formula cheatsheet”

  • Line y=mx+by=mx+by=mx+b: set y=0y=0y=0, solve 0=mx+b0=mx+b0=mx+b ⇒ x=−b/mx=-b/mx=−b/m.
  • Line Ax+By+C=0Ax+By+C=0Ax+By+C=0: set y=0y=0y=0, solve Ax+C=0Ax+C=0Ax+C=0 ⇒ x=−C/Ax=-C/Ax=−C/A.
  • Any function y=f(x)y=f(x)y=f(x): solve f(x)=0f(x)=0f(x)=0 for xxx; those solutions are the x‑intercepts.

Quick story‑style example

Imagine a ball rolling along a curved track drawn on a graph. At some point, it hits the floor level—that’s the x‑axis. Wherever it touches the floor, the height yyy is zero. If you know the rule of the track (the equation), you just tell the rule “be at floor level” by putting y=0y=0y=0 and then solve to see where along the floor (which xxx) that happens.

TL;DR:
To find an x‑intercept, set y=0y=0y=0 and solve for xxx (or just look where the graph crosses the x‑axis and read the xxx value).

Information gathered from public forums or data available on the internet and portrayed here.