why does b stand for y intercept
I ve been teaching and tutoring the slope intercept form of the equation of a line for 25 years. That whole time I ve wondered, Why is the y-intercept called
b? Just today I discovered the answer! We teach it backwards. As is typical, we teach things in reverse of how we create them. The standard (or general) form of a linear function is f(x) = ax + b f(x) notation isn t really needed when you first learn to graph. We use y instead. So we have y = ax + b x a ) is the slope or the measurement of how much the line tilts. m. So we change the basic equation to include the slope notationБ y = mx + b b here is also exactly where the line smacks into the y-axis AKA the y-intercept. But there s no fancy letter for that, so we just leave it as b. And we present that first! y = mx + b m b = y-intercept. But we don t bother to tell them why we re using such crazy letters! But now you have it. That s the answer to the question, Why is the y-intercept called b? m! ) This post may contain affiliate links. When you use them, you support us so we can continue to provide free content!
Back when we were first graphing straight lines, we saw that the slope of a given line measures how much the value of y x changes. For instance, consider this line: m. This means that, starting at any point on this line, we can get to another point on the line by going up units. But (and this is the useful thing) we could also view this slope as a fraction over This tells us, in practical terms, that, for every one unit that the x -variable increases (that is, moves over to the right), the y -variable increases (that is, goes up) by three-fifths of a unit. While this doesn't necessarily graph as easily as three up and five over, it can be a more useful way of viewing things when we're doing word problems or considering real-world models. Very often, linear-equation word problems deal with changes over the course of time; the equations will deal with how much something (represented by the value on the vertical axis) changes as time (represented on the horizontal axis) passes. An exercise might, say, talk about how the population grows, year on year, in a certain city, assuming that the population increases by a certain fixed amount every year.
For every year that passes (that is, for every increase of along the horizontal axis), the population would increase (that is, move up along the vertical axis) by that fixed amount. x, the corresponding y y -intercept. In the particular context of word problems, the y -intercept (that is, the point when x ) also refers to the starting value. For a time-based exercise, this will be the value when you started taking your reading or when you started tracking the time and its related changes. In the example from above, the y -intercept would be the population when the sociologists started keeping track of the population. If they started taking their measurements or doing their calculations from a base year of 1997, then x would correspond to the year 1997, and the y -intercept would correspond to the population in 1997. The following are a few examples to help illustrate how this works. The average lifespan of American women has been tracked, and the model for the data is y = 0. 2 t, where t corresponds to 1960.
Explain the meaning of the slope and the y -intercept. v = 128 32 t, where v t is the number of seconds after the ball is thrown. With what initial velocity was the ball thrown? What is the meaning of the slope? Fisherman in the Finger Lakes Region have been recording the dead fish they encounter while fishing in the region. The Department of Environmental Conservation monitors the pollution index for the Finger Lakes Region. The model for the number of fish deaths y x y = 9. 607 x + 111. 958. What is the meaning of the slope? What is the meaning of the y -intercept? Word problems with linear equations (that is, with straight-line models) almost always work this way: the slope is the rate of change, and the y -intercept is the starting value. (I can't, off the top of my head, think of any instance in which this would not be the case. ) The main difficulty is usually in interpreting the horizontal variable, especially when that variable is keyed to a certain year. Always make sure that you're clear on the definitions of the variables, and you should be fine.
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why do they use the letter m for slope
why is the y intercept called b