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