Rational and Exponential Functions

Rational function
1. A function that is the ratio of two polynomials
  1. This is a rational function because the denominator is divided by the numerator
Horizontal & Vertical Asymptotes of Rational Functions
1. Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function
  1. 1 is the vertical asymptote of the graph because in the donominator x minus 1 is zero so the denominator cannot be divided by the numerator which is one
    1. Also on in the graph the graph approaches 1 on the x-axis but, does not touch it
2. Horizontal asymptotes are where the graph approaches a value across the y-axis but does not touch it
  1. The horizontal asymptote of this graph is y=2
  2. You can look at the domain to determine the horizontal asymptote
    1. For example if the domain is all x-values other than ± 3/2, and the two vertical asymptotes are at x = ± 3/2.
Exponential function
1. function in which an independent variable appears in one of the exponents
  1. y=ab^x
Horizontal Asymptotes of Exponential Functions
1. The horizontal asymptote is where the graph approaches a value across the y-axis but does not touch it
  1. The number that is being added in the equation is the horizontal asymptote of the equations above
End behavior of a graph with asymptotes
1. the end is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity
  1. The function on the right side increases and approaches infinity
    1. The function on the left side decreases and approaches negative infinity
Y-intercept of an exponential function
1. The Y-intercept of an exponential function iswhere the function crosses the y-axis
  1. The x value of a Y-intercept will always be zero so when x is zero the function crosses the y axis
    1. The Y-intercept of this graph is (0,1)
Exponential growth function
1. The exponential growth function is k>0
  1. In this function k is known as the growth factor
    1. Growth factor is the factor by which a number multiplies itself over time
Exponential decay function
1. The exponential decay function is k<0
  1. In this function k is known as the decay factor
    1. Decay factor is the factor by which a number divides itself over time
Growth factor of an exponential function
1. When a > 0 and b > 1, the function models growth.
  1. b is the growth factor and a is the initial amount
Decay factor of an exponential function
1. When a > 0 and 0 < b < 1, the function models decay
  1. b is the growth factor and a is the initial amount
Compound Interest Formula
1. A represents amount of money accumulated after n years, including interest.
2. P represents principal amount (the initial amount you borrow or deposit)
3. r represents annual rate of interest (as a decimal)
4. n represents number of times the interest is compounded per year
5. t represents number of years the amount is deposited or borrowed for.
Continuous Compounding
1. A = amount after time t
  1. P = principal amount (initial investment)
    1. r = annual interest rate (as a decimal)
      1. t = number of years
2. The continuous compound formula can be used to find the balance at the bank

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Rational and Exponential Functions

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