Exponential Functions

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Exponential Functions

Zero and negative exponents

Any fractional base raised to a negative exponent is equal to its reciprocal of the base raised to the same positive exponent

For example: (a/b^-m)= b/a^m

Any base raised to a negative exponent is equal to the reciprocal of the base raised to the same positive exponent

For example: 2^-4/3^-4=3^4/2^4

Any value raised to the power of zero is equal to one

For example: 7^0=1

Compound interest

There are many different compounding periods. They include annually (1 time/year), semi-annually (2 times/year), quarterly (four times/year), monthly (12 times/year), semi-monthly (24 times/year), weekly (52 times/year), bi-weekly (26 times/year), and daily (365 times/year).

Compound interest: with compound interest, the interest you earn is based on the principal and the interest from the previous period.

For example: $5000 is invested at 8% compounded quarterly. By looking at this statement you know that the principal amount is $5000, the interest is 0.02 (you divide the given interest by four because it is compounded quarterly), and the n value is 20 because it is compounded quarterly for five years (4x5=20). You solve this by substituting all the values into the formula, and you should get a=$7429.74.

The formula for compound interest is A=P(1+i)^n

A is the amount of principal and interest, P is the present value of the investment or debt, i is the rate of interest per period, and t is the total number of periods

Your principal amount for the next period is the same as your final amount for the previous period.

Simple Interest: With simple interest the interest you earn is based only on the principal.

The formula for simple interest is: I=prt

I stands for simple interest, p stands for principal amount, r stands for annual rate, and t is time in years.

Power of power

For example: (a^3)^2= (a^3)(a^3)=a^6

When raising a power to another power the exponents get multiplied

When raising an exponent to a power it is just like raising a regular number to a power

Properties of exponential functions

If b=1, then y=b^x is a straight line

If b<1, then y=b^x is a straight line

If b>1, then the y=b^x is continuously decreasing

The x-axis is a horizontal asymptote

Asymptote is a line/value that you get closer and closer to but never reach. The line of the graph gets closer and closer to the x-axis but never touches or crosses it.

The domain is {x|xER}, and the range is {y|y>0 yER}

The function of y=b^x where b>0 defines an exponential function

Rational Exponents

Examples of powers with rational exponents: x^n/m

n is a natural number, x>0 when n is a even number, if x<0, n must be an odd number.

A rational number is a number that can be expressed as the quotient of two integers (a fraction) where the divisor is not zero.

For example: 0.75, -3/4, -2.

Powers and exponents

When dividing exponents with the same base, you keep the same base and subtract the exponents.

For example: 3^6/3^2=3^4

When you have a negative exponent, you put one over the exponent it turns positive.

For example: 3^-2=1/3^2=1/9

When multiplying exponents with the same base, you keep the base and add the exponent.

For example: 2^2 x 2^2=2^4

a^n is an exponential expression, where a is the base and n is the exponent.