What does the x intercept represent?

Answer 1

The Simple Answer

This is a special term; it is short for, "the point where an equation intercepts the x axis".

(Technical note. The x axis is the line on which you mark all the possible x values, and the y axis is the line on which you mark all the y values. These are always drawn so that they cross at 0,0. That is arbitrary; there is no (mathematical) reason to draw the line representing all the x values at the point where y=0, and there is no (mathematical) reason to draw the line representing all the y values at the point where x=0. Still, it makes very good sense to always draw them in the same place, and putting them at 0 also makes very good sense. It is thus that "x axis" means "y=0" and "y axis" means "x=0", by convention.)

In turn, the "x axis" is the horizontal line at y=0.

Thus, if you had (for instance) the equation "y = x + 2", and they ask for the "x intercept", that means that you have to work out "the point where this equation intercepts the x axis", which in turn means "the point, in this equation, where y=0".

To work this out, you feed "y=0" into the equation, giving "0 = x + 2". The solution is "x = -2", because "-2 + 2 = 0". (How to work this out is a separate question; you use algebra to turn it into "x = …".)

More Advanced Stuff

In maths, lines are represented as "functions" (such as "y = x + 2" ). They are called this because y is a function of x. That means that you use an equation (such as "y = x + 2" ) by replacing "x" with a value (say 3), and working out what y is. (For x=3, y=5, in this instance.)

The reason I mention this is that you can never have more than one "y" answer for any given value for "x"… but you can have some y values appear more than once (at different x values). In the case of a simple parabola - "y = x^2" - there are two points where y=4 - at x=2 and x=-2. Conversely, you do not ever have two different y values at the same x value - not in this equation, nor in any other.

(If you were to draw a line - a circle, for instance - that had more than one y value for a given x… you could not write a function for this. Yes, it does have an equation - "x^2 + y^2 = 3", for instance… but if you convert this into a function (that is, in the form "y = …", with y as a function of x), you lose the points at which y is negative, and you get only half of the circle (the top half).)

Getting to the point finally… the y intercept [which is not what the question is about] is about the y axis; it is where the line crosses the y axis. The y axis is at x=0. So… "What is the y intercept?" means "What is the y value when x=0?" To work that out, just make x = 0 in the equation, and… presto! There is only one answer, and the equation will tell you what it is.

The x intercept is about the x axis. This is at y=0. In contrast with the above, there can be more than one answer. The way to work it out is to make x the subject. That, of course, will give you x as a function of y, meaning that now you can have only one answer - since it is a function, there can be only one x value for y=0.

The point of this "more advanced" section is that, for harder equations, working out the "x intercept" can involve knowing about when an equation might represent more than one "x intercept" value, and how to do the working out so as to get more than one answer.

(A sine curve, for instance, crosses the x axis repeatedly - twice for each cycle.)

A typical "x intercept" question will have a "polynomial" equation. That means one with, for instance, an "x", an "x^2" and an "x^3". (Terminology: Each of those is called a "term". The numbers are called "exponents". The highest exponent is the most significant, and the term for the highest exponent is "order"; in this case it is a 3, so this would be called a "third-order polynomial" ("3" becomes "third").)

Working backwards…

Say we have "(x-3)(x+4)=0". Since we are multiplying, the answer will = 0 whenever any one set of parentheses = 0. Thus, there are two answers here (since there are two things in parentheses). We will get 0 when x=3 (since 3-3=0) and when x=-4 (since -4+4=0).

As for the polynomial: "(x-3)(x+4)=0" comes out as "x^2 +1x -12". The quick way to work this one out (forwards) has to do with the fact that -3+4=1, and -3x4=-12. (The mathematical way is called "completing the square", and, no, I am not going to explain it.)