How is an exponential function different from a polynomial function? Examples?
polynomials may have "wiggles" in them, and have either
(a) a max or a min
(b) go from -ā to +ā
pure exponentials (y=a*b^x) have a horizontal asymptote at y=0
are always positive or negative, depending on the sign of a
at some point they grow faster than any polynomial, no matter how high a degree
An exponential function and a polynomial function are both common types of mathematical functions, but they have different structures. Here are the key differences between the two:
Exponential function:
- An exponential function has the general form: f(x) = a * b^x, where 'a' and 'b' are constant values.
- The variable 'x' is an exponent, meaning it appears as a power.
- The base 'b' is typically a positive number greater than 1.
- As 'x' increases or decreases, the function grows or decays exponentially.
- Example: f(x) = 2^x, f(x) = 3 * (1.5)^x.
Polynomial function:
- A polynomial function has the general form: f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_1 * x + a_0, where 'n' is a non-negative integer and 'a' coefficients are constant values.
- The variable 'x' appears as multiple terms raised to different powers.
- A polynomial can have any degree, determined by the highest power of 'x'.
- Polynomial functions can have positive, negative, or zero exponents.
- Example: f(x) = 5x^3 + 2x^2 - 3x + 1, f(x) = 2x^5 + 7x^2.
In summary, exponential functions involve a base raised to the power of 'x', while polynomial functions contain terms with 'x' raised to different powers.
An exponential function and a polynomial function are quite different from each other.
A polynomial function is a mathematical function consisting of one or more terms, where each term has variables raised to non-negative integer powers. In other words, a polynomial function is a sum of terms, where each term is a constant multiplied by a variable raised to a whole number power. For example, f(x) = 2x^3 + 5x^2 - 3x + 7 is a polynomial function of degree 3.
On the other hand, an exponential function is a mathematical function of the form f(x) = a^x, where a is a constant base and x can be any real number. The input variable x is the exponent, and the base a determines how the function grows or decays. For example, f(x) = 2^x and g(x) = e^x (where e is the base of the natural logarithm) are examples of exponential functions.
One key difference is in how the functions grow as the input variable increases. In a polynomial function, the function may increase or decrease, but it will do so at a relatively steady rate. In an exponential function, the function typically grows or decays very rapidly as the input variable increases or decreases.
For example, let's compare the polynomial function f(x) = x^2 and the exponential function g(x) = 2^x. If we calculate some values, we can observe the differences:
f(0) = 0^2 = 0
f(1) = 1^2 = 1
f(2) = 2^2 = 4
f(3) = 3^2 = 9
So, as x increases by 1, the output of the polynomial function increases by successive odd numbers, i.e., 1, 3, 5, and so on.
Now, let's look at the exponential function:
g(0) = 2^0 = 1
g(1) = 2^1 = 2
g(2) = 2^2 = 4
g(3) = 2^3 = 8
As x increases by 1, the output of the exponential function doubles. The exponential function grows much faster compared to the polynomial function.
In summary, the main difference between an exponential and polynomial functions lies in their growth rates. Polynomial functions grow at a relatively steady rate as the input variable changes, while exponential functions grow or decay rapidly.