Slope intercept form

Slope intercept form is y=mx + b where m is the slope and b is the y intercept.

Why does the equation use the y intercept instead of the x intercept. Can anyone explain?

You need to write y = m(x - a) if you want to see the x-intercept.

You can do it either way, of course, but one of the things it is desirable to teach students is the concept of a function.

Think of a function as a black box. If you enter ‘x’ into this box, you always one value out. Let’s call it f(x). Further, it’s always the same output as long as you enter the same ‘x’ each time.

For example, let’s say we had our black box. Call it f. And let’s say we put in some values of x to test our function f:

Input x Output of f
0 0
1 2
5 10
100 200
8034 16068

Whenever you put in x=1, you always get 2 out. Never another value and never just nothing.

We can reasonably infer that our function f takes the input x and just doubles it. In other words, f maps x to 2x, or in more traditional notation: f(x) = 2x.

Slope-intercept form is usually taught in the context of a particular kind of function - linear functions, which means functions are outputs that depend only on some factor of x to the first power, possibly with a constant shift, or of the form:

f(x) = mx + b
Or even a

When we write y=m*x+b, we are being maybe slightly sloppy notationally, though this sometimes makes things conceptionally simpler for students. Really, we mean that we have some function of x, which we are denoting using y, i.e. y = f(x). The different ‘forms’ are really ways to help students figure out what linear functions look like if only given a set of pairs of values, x and corresponding f(x).

Ok, so why not the x-intercept? Let’s now say we have a more complicated function. Say we have a function f such that:

f(x) = x^2 - 4

This function has a single y-intercept (all functions have a single y-intercept*). That y-intercept is -4. As for x-intercepts, this function actually has 2, not just 1, x-intercepts at x=2 and x=-2.

Ok, let’s flip one sign and see what we get:

f(x) = x^2 + 4

This function still has a single y-intercept (again, all functions have a single y-intercept*). That y-intercept is 4. But this particular function has zero x-intercepts**.

And that’s why we focus on y-intercepts more than x-intercepts. As long as the function is defined for x=0, it will have a y-intercept. But several functions don’t have any x-intercepts.

The existence and value of those x-intercepts is actually a rather important branch of algebra. When we talk about ‘solving equations’, we basically mean finding all those x-intercepts, if they exist. And the existence of such solutions in the first place is itself an involved area of study that occupied the lives of several mathematicians over the centuries.

well, as long as the domain of support is defined over x at 0, but that’s getting a bit too advanced for an already very long post

well, it has no x-intercepts unless the function is defined over complex numbers, but again that’s getting a bit advanced for this post

Like DPRK: said, you can use algebra to rearrange the formula however you need to.

We tend to call our first variable ‘x’ and are next ones, ‘y’ and then ‘z’ because of some random convention. I think that doiesn’t hold true for all languages, however.

There’s another reason for the way we arrange, ‘x’ and ‘y’, or abscissa and ordinate, or "the horizontal’ and ‘the vertical’ – except that last one brings out the pendants to remind us they’re both horizontal (in a book on our laps) or both vertical (on chalkboard or computer screen.) We use ‘x’ for the independent variable, ‘y’ is the ‘dependent’ variable.

Dude, you will get it by simply rearranging the equation.

This is a good point, but I think missing a critical element. There is a convention that says you print a graph with the dependent variable plotted vertically and the independent variable plotted horizontally. I got the chance to speak to some expert on graphing once, and I asked them about this, and it seems this is an unspoken and assumed thing that nobody gets creative about. For example, if a dependent variable represents a fundamentally horizontal phenomenon such as the position of a pendulum bob, it still confuses absolutely everybody to actually use the horizontal axis to represent it.

Eh, I don’t think that makes any difference for the algebraic form. If we used the vertical axis for the independent variable (which is actually occasionally done, especially when the independent variable is time), then we’d just speak of a horizontal-line test instead of a vertical-line test. But the algebraic form would still look the same.

Note that in a purely geometric context the variables x and y play purely symmetric roles and worrying about functions of one in terms of the other may not be as important as more generally considering systems of equations; for example, x[sup]2[/sup] + y[sup]2[/sup] = 1 describes a circle. To calculate the intersection with the x-axis or the y-axis, adjoin another equation y = 0 (or x = 0).

y = mx + b gives y as a (linear) function of x, as Great Antibob explains.

And in that case, the y-intercept often has a special significance, as a “starting value” for the function.

For example, if your equation is a formula for linear depreciation (how much a piece of equipment is worth from year to year, as it gets older and loses value), the y-intercept would be how much it’s worth starting out—its value when new.

If your equation is a linear cost function (cost to make x number of T-shirts or coloring books or Chevy Malibus), the y-intercept represents the fixed cost, or overhead—your expenses that stay the same no matter how many T-shirts you make, before you start adding on the cost per T-shirt.