Intercepts describe where a graph meets the coordinate axes. Intersections describe where two graphs have the same coordinates. These points are useful for solving equations, comparing models, and defining shaded regions.
X-intercepts
An x-intercept occurs where y=0. For a function y=f(x), solve f(x)=0.
Example: f(x)=x^2-4. Solving x^2-4=0 gives x=-2 and x=2, so the x-intercepts are (-2,0) and (2,0).
Y-intercepts
A y-intercept occurs where x=0. Evaluate f(0).
For f(x)=x^2-4, f(0)=-4, so the y-intercept is (0,-4).
Intersections of two graphs
Two graphs intersect where their y-values are equal. To find the intersections of y=x^2-4 and y=x+2, solve:
x^2 - 4 = x + 2 x^2 - x - 6 = 0 (x - 3)(x + 2) = 0
The x-values are 3 and -2. Substituting into y=x+2 gives the intersection points (3,5) and (-2,0).
Use the graph to locate, then calculate to confirm
Graph both functions with different colors. The visual intersection provides a fast estimate and helps detect whether there are zero, one, or several solutions. Algebra then provides exact values when available.
Tangent intersections
Two curves can touch at one point without crossing. A graph may make the contact difficult to see, especially at a coarse scale. Zoom in and compare derivatives if you need to determine whether the curves are tangent.
Intersections outside the visible window
Do not conclude that two graphs never meet merely because no intersection appears on screen. Change the viewing range or analyze the equation. Exponential, logarithmic, and high-degree functions can intersect far from the origin.
Approximate solutions
Some equations do not have a simple exact solution. In those cases, use the graph and analysis tools to estimate the coordinates. Report an appropriate number of decimal places and make it clear that the result is approximate.
Common mistakes
- Calling every point on the x-axis a y-intercept.
- Finding x-values but forgetting to calculate the corresponding y-values.
- Rounding too early.
- Missing a repeated root where the graph touches the axis.
- Using a viewing window that hides an intersection.
Graph the worked example and inspect both intersections →
To understand how formulas move and reshape graphs, continue with function transformations.