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Rational and Exponential Functions

1 Rational function

1.1 A function that is the ratio of two polynomials

1.1.1 This is a rational function because the denominator is divided by the numerator

2 Horizontal & Vertical Asymptotes of Rational Functions

2.1 Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a
rational function

2.1.1 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

2.1.1.1 Also on in the graph the graph approaches 1 on the x-axis but, does not touch it

2.2 Horizontal asymptotes are where the graph approaches a value across the y-axis but does not touch it

2.2.1 The horizontal asymptote of this graph is y=2

2.2.2 You can look at the domain to determine the horizontal asymptote

2.2.2.1 For example if the domain is all x-values other than ± 3/2, and the two vertical asymptotes are at x =
± 3/2.

3 Exponential function

3.1 function in which an independent variable appears in one of the exponents

3.1.1 y=ab^x

4 Horizontal Asymptotes of Exponential Functions

4.1 The horizontal asymptote is where the graph approaches a value across the y-axis but does not touch it

4.1.1 The number that is being added in the equation is the horizontal asymptote of the equations above

5 End behavior of a graph with asymptotes

5.1 the end is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity

5.1.1 The function on the right side increases and approaches infinity

5.1.1.1 The function on the left side decreases and approaches negative infinity

6 Y-intercept of an exponential function

6.1 The Y-intercept of an exponential function iswhere the function crosses the y-axis

6.1.1 The x value of a Y-intercept will always be zero so when x is zero the function crosses the y axis

6.1.1.1 The Y-intercept of this graph is (0,1)

7 Exponential growth function

7.1 The exponential growth function is k>0

7.1.1 In this function k is known as the growth factor

7.1.1.1 Growth factor is the factor by which a number multiplies itself over time

8 Exponential decay function

8.1 The exponential decay function is k<0

8.1.1 In this function k is known as the decay factor

8.1.1.1 Decay factor is the factor by which a number divides itself over time

9 Growth factor of an exponential function

9.1 When a > 0 and b > 1, the function models growth.

9.1.1 b is the growth factor and a is the initial amount

10 Decay factor of an exponential function

10.1 When a > 0 and 0 < b < 1, the function models decay

10.1.1 b is the growth factor and a is the initial amount

11 Compound Interest Formula

11.1 A represents amount of money accumulated after n years, including interest.

11.2 P represents principal amount (the initial amount you borrow or deposit)

11.3 r represents annual rate of interest (as a decimal)

11.4 n represents number of times the interest is compounded per year

11.5 t represents number of years the amount is deposited or borrowed for.

12 Continuous Compounding

12.1 A = amount after time t

12.1.1 P = principal amount (initial investment)

12.1.1.1 r = annual interest rate (as a decimal)

12.1.1.1.1 t = number of years

12.2 The continuous compound formula can be used to find the balance at the bank

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