Thread: How can you tell the difference between linear and exponential function ?

are these linear or exponential?? (please explain why)

y=x-3
y=3 to the power of x
y=-2(2)to the power of x

A linear function of x has the form a*x+b.

An exponential function of x has the form a*(b^x)
... where a and b are numbers, "^" means "raised to the power", and "*" is multiplication.

You can remember this as "exponential means x is the exponent"

So:
y = x - 3 ..... linear (a=1, b=-3)
y = 3^x ..... exponential (x is in the exponent)
y = 2*(2^x) ..... exponential (x is in the exponent)

It's a little more flexible than that, actually. Take a look at:

3 * (10 ^ (2x + 4))

There, the exponent isn't x, but a linear function (2x+4) of x instead. However, use the rules of exponents:

3 * (10 ^ (2x + 4)) = 3 * 10^(2x) * 10^4
= (3 * 10^4) * 10^(2x)
= (3 * 10^4) * (10^2)^x

So 3 * (10 ^ (2x + 4)) converts to the form a*(b^x), where a=(3* 10^4) and b=10^2. So a function is still exponential in the a*b^x sense, even if the exponent is a linear function of x.

Final note: Don't confuse 2^x with x^2. The first is exponential, but the second is quadratic. If you plot them, you'll see that 2^x grow MUCH faster than x^2.