الرياضيات

What Is the Y-Intercept?

The value of y when x is zero. Read −4 straight off y = 3x − 4, then use the substitute-zero method for equations and curves that are not in that form.

y-intercept

The point where a graph crosses the y-axis, which is the value of y produced when x is zero.

أين يقابله الطلاب: Grade 7 or 8 mathematics, alongside y = mx + c, and again whenever curves are sketched at GCSE and IGCSE.

الإجابة باختصار
The y-intercept is where a graph crosses the y-axis, which is the value of y when x = 0. For y = 3x − 4 it is −4, read straight off the constant. For any other equation, substituting x = 0 and working out y gives it, whether the graph is a line or a curve.

مثال

y = 3x − 4 → y = 3(0) − 4 = −4 → (0, −4)

Putting zero in for x removes the 3x term entirely, and whatever is left is the intercept. That is the whole reason c can be read off y = mx + c without any working: at x = 0 the mx part contributes nothing, so the constant is the height at which the line crosses. The same substitution works on anything. For y = x² − 5x + 6 it gives 6, and for y = 2ˣ it gives 1, because anything to the power zero is 1.

When you can simply read it

In y = mx + c the constant c is the y-intercept, and there is a reason rather than a rule behind it. Setting x to zero wipes out the mx term and leaves y = c, so the constant is the height of the crossing point.

y = 3x − 4 crosses at −4. y = 2x crosses at 0, which is the same as saying it passes through the origin. y = 7 − x crosses at 7, and the fact that the constant is written first changes nothing.

The method that never fails

The moment an equation is not in y = mx + c form, reading numbers off the page stops working, and this is where marks go. Take 2x + 5y = 20. The intercept is not 20. Substituting x = 0 gives 5y = 20, so y = 4 and the graph crosses at (0, 4).

Substituting zero is worth treating as the actual definition, with the y = mx + c shortcut as a special case of it. It works on curves without modification: y = x² − 5x + 6 crosses at 6, y = (x − 3)(x + 5) crosses at −15, and a reciprocal graph such as y = 1/x has no y-intercept at all, because x = 0 is not a value it accepts.

Finding it when x = 0 is not on the page

Tables of values often start at x = 1, and coordinate questions hand you two points somewhere out in the middle of the grid. In both cases the intercept is still reachable by stepping backwards along the line.

Suppose a line has gradient 3 and passes through (2, 5). Going from x = 2 to x = 0 means moving two steps left, and each step left drops y by 3, so the intercept is 5 − 6 = −1 and the equation is y = 3x − 1. Substituting the point into y = 3x + c does the same arithmetic more formally: 5 = 6 + c, so c = −1.

أسئلة شائعة

Is the y-intercept a number or a coordinate?

Both usages are common, and the question tells you which it wants. Asked for the point where the graph crosses the y-axis, write (0, −4). Asked for the y-intercept or the value of c, write −4. When you are unsure, giving the coordinate is safer, because it contains the number as well.

Can a graph have more than one y-intercept?

A function cannot. Crossing the y-axis twice would mean x = 0 producing two different outputs, which no function is allowed to do. Some graphs that are not functions can — a circle drawn on axes may cut the y-axis at two heights. A vertical line such as x = 3 has none at all.