ASAP Question EASY!!!?

Question

is "b" in the formula Y=ab^x the slope(rate of increase)?

Answer 1

No, b in that formula is not the slope. It is not a straight line equation.

You have to go to calculus to realize that every point on that curve has its own slope, which can be calculated with the theories of calculus. For these kinds of curves, the slope at any point is proportional to y itself! It's called a "growth curve" where your increase as you move along is proportional to the value of the curve at that point. Just like the increase in human population every year depends on how much of that population you already have. Or how much interest you'll get every year from money in the bank. The percentage growth is a constant, but in real terms the increases can get monstrously large, since the growth itself can grow some more!

The slope in calculus is represented as "dy/dx" (what the infinitesimal change in y is at that point with every change in x)

if y = ab^x
dy/dx = ab^x * ln b

ln b ("the natural logarithm of b") is a kind of correction function. When they looked at growth curves of the form b^x, they found out that there was a certain value of b that you could choose where the slope of the curve would be equal to EXACTLY b^x itself. They found that happened when b was equal to a number called e = 2.718 281 828 459... [This number seems to start with a pattern of 1828 repeated twice, but it is NOT a repeating decimal.]

ln e = 1 (no correction, in a sense)
ln e^2 = 2, etc.

[Just like the logarithms of the powers of 10 equal their exponents.]

If you don't know calculus yet, don't sweat it. Differential calculus allows you to find the slope of any continuous function, so if you study hard in math, you will have awesome tremendous powers to solve problems having to do with rates of change! It will almost make up for the first 10 years of boring stuff!

Answer 2

NO.

If you have a formula where y=ax^2, you don't have a slope.

The "SLOPE" as defined in basic math courses are linear. Which is, slope 2 means for every 1 increase in X, you have 2 increase in Y.

If your formula raises a, then b raised to power of x, that will make, for every increase in X, Y will increase exponentially.

Simply put, you do NOT have a slope.