Zeros, y-Intercept and Intersections of Graphs
Zeros, the y-intercept and intersection points are the special points where a function graph meets the x-axis, the y-axis or another graph. In brief: a zero of f is an x with f(x) = 0, the y-intercept is the point (0, f(0)), and the intersection of two graphs is found by solving f(x) = g(x). For f(x) = 2x − 6 the zero is x = 3 and the y-intercept is (0, −6).
These points give a quick description of a graph and often carry real meaning: the break-even point of a business, the moment a ball lands, the time when two cars are at the same position. This guide shows how to find each of them algebraically and graphically for lines, quadratics and other functions.
What is a zero of a function?
A zero, or root, of a function f is a number x for which f(x) = 0. On the graph it is a point where the curve touches or crosses the x-axis, and such a point is called an x-intercept. A function can have no zero, one zero or many. The function f(x) = x2 + 1 has none, f(x) = x2 has one (x = 0) and f(x) = sin x has infinitely many.
To find the zeros, solve the equation f(x) = 0. For a line, this is a linear equation. For a quadratic, use factoring or the quadratic formula.
What is the y-intercept?
The y-intercept is the point where the graph meets the y-axis, which you can also read from a table or a graph. On the y-axis x = 0, so the y-intercept is (0, f(0)). A function has at most one y-intercept, because a function assigns only one output to x = 0. If 0 is not in the domain, as for f(x) = 1/x, there is no y-intercept.
For f(x) = 3x2 − 5x + 4 the y-intercept is (0, 4): simply the constant term. For a line y = mx + b, it is (0, b).
How do you find where two graphs intersect?
- Set the two function rules equal: f(x) = g(x).
- Solve for x. Each solution is the x-coordinate of an intersection point.
- Substitute each x into f or g to get the y-coordinate.
- Check using the other function.
The number of intersection points equals the number of solutions of f(x) = g(x). The graphs may have none, as with y = x and y = x + 1, which are parallel lines.
Worked examples
Example 1: line
f(x) = 2x − 6. Zero: 2x − 6 = 0, so x = 3. The y-intercept is f(0) = −6, so (0, −6).
Example 2: quadratic
f(x) = x2 − x − 6 = (x − 3)(x + 2). The zeros are x = 3 and x = −2. The y-intercept is (0, −6).
Example 3: intersection of a line and a parabola
Find the intersections of f(x) = x2 and g(x) = x + 2. Solve x2 = x + 2, so x2 − x − 2 = 0, which factors as (x − 2)(x + 1) = 0. The x-values are 2 and −1. The points are (2, 4) and (−1, 1). Check with g: g(2) = 4 and g(−1) = 1 ✓.
Example 4: break-even point
A stall has costs C(x) = 40 + 2x and income R(x) = 5x for x items. Break-even occurs when 40 + 2x = 5x, so x = 40/3 ≈ 13.3. Whole items are needed, so the stall makes a profit from the 14th item on. Check: C(14) = 68 and R(14) = 70.
Zeros and intercepts of common functions
| Function | Zeros | y-intercept |
|---|---|---|
| f(x) = x − 4 | x = 4 | (0, −4) |
| f(x) = x2 − 9 | x = −3 and x = 3 | (0, −9) |
| f(x) = (x − 1)(x − 2)(x − 3) | x = 1, 2, 3 | (0, −6) |
| f(x) = √x − 2 | x = 4 | (0, −2) |
| f(x) = 1/x | none | none |
| f(x) = 2x | none | (0, 1) |
| f(x) = x2 + 4 | none (real) | (0, 4) |
Where do these ideas come from?
Finding zeros of polynomials was a main goal of algebra from its beginning, from Babylonian quadratic problems to Cardano’s solution of cubic equations in the sixteenth century. Intersection of a line and a curve appeared in the Greek study of conic sections. With the introduction of coordinates in the seventeenth century, both problems became questions about equations, and the term “intercept” came into use in the study of graphs.
Where are these points used today?
The zero of a height function h(t) = 20t − 5t2 gives the time when a thrown ball returns to the ground: t = 0 and t = 4 seconds. Intersections show when two plans cost the same, when two moving objects meet or when supply equals demand. The zero of a profit function marks the break-even point, and a y-intercept often represents a fixed starting value, such as a fee paid before any use.
Which mistakes are common?
Mixing up the two intercepts
For the y-intercept set x = 0. For the x-intercept set y = 0.
Forgetting a second zero
Quadratics can have two zeros. After dividing by x, you can lose the solution x = 0: from x2 = 3x, write x(x − 3) = 0, not x = 3.
Stopping at the x-coordinate
Intersection points are pairs. Always compute the y-value.
Assuming a graph always crosses the axis
Some graphs only touch the x-axis, like y = x2 at the origin, and others never meet it.