Exponential Functions
b [A(t)= a x b^t]
(1 + r) or (1 - r) depending on growth or decay
ANYTHING TO THE 0
1
Horizontal Asymptote (Exponent Graphs)
A line across the y axis that is worked toward but NEVER TOUCHED OR GONE PAST; Always draw DOTTED LINE; Written as Y=#
Compound Interest Formula
A=P(1+r/n)^nt
ALWAYS WRITE PERCENT AS A DECIMAL
ALWAYS WRITE PERCENT AS A DECIMAL
A (Compound Interest Formula)
Amount in account after t years
If you have a logarithm WITH THE SAME BASE on both sides
Cancel the logs by crossing them out if they are on both sides
Solving log equations if there is only one log
Convert the log to an exponent and solve
A(t)= a x b^t
EVERYTHING IS THE SAME AS EXPONENTIAL GROWTH FORMULA BESIDES B
F(x)=ab^(x-h) + k
Exponential function formula for transformations
Exponential Growth Function
F(x)=b^x; b > 1; HORIZONTAL ASYMPTOTE established by the outside parenthesis transformation (y-axis); INFINITE BOTH WAYS ON THE X AXIS
Exponential Decay Function
F(x)=b^x; b is in between 0 and 1; HORIZONTAL ASYMPTOTE established by the outside parenthesis transformation (y axis)]; INFINITE BOTH WAYS ON THE X AXIS
Graphing Log Functions
Find the asymptote, find an easy x value to make an easy answer for the log, graph your points
Solving Log Inequalities
Find where each log is undefined, solve the actual inequality, USE ALL THE ANSWERS TO MAKE A NUMBER LINE, then test values to make a final solution
Graphing Exponential Functions
Graph the horizontal asymptote using k, Use the exponent to find the x values based on the transformation, Make a table with x and y values, and plot and point the graph
1/anything to the -1 power
IS ALWAYS WHATEVER THE DENOMINATOR IS
P (Compound Interest Formula)
Initial/Principal invested amount
r (Compound Interest Formula)
Interest rate (decimal)
Vertical Asymptote (Log Graphs)
Inverse of the exponent graphs; Written as x=#; Shifted inside the parenthesis by H; GOES UP AND DOWN
If asked to write an equation the represents an exponential situation
JUST FILL IN THE EQUATION WITH PROVIDED NUMBERS ; DO NOT DO ANY OTHER MATH AFTER
Logarithms
Just Exponents; Inverse of exponent form; X IS GREATER THAN 0 AND B IS GREATER THAN 0
When Doing Log Inequalities Where the Variable Isn't The Logicand
Just solve it normally, disregard the log inequality rules
Log Language
Log base (whatever) of x
F(x)= a (log) base (x-h) + k
Log graph formula
Finding Where the Log is Undefined (Log Inequalities)
MAKE THE LOGICAND GREATER THAN 0
Testing for Values (Log Inequalities)
Make a number line and find the areas the meet ALL THE CONDITIONS YOU HAVE SET FOR X
Expressing Things In Terms Of Common Logs
Make an equation and set whatever isn't in common log form equal to LOG BASE 10 TO THE X
Opposite
Multiply whatever it is by negative 1
Checking Log Equations
Must put your answer for the variables into logicands to make sure the total logicand is positive
n (Compound Interest Formula)
Number of times compounded per year
t (Compound Interest Formula)
Number of years
r [Exponential Decay Formula)
Percent decrease per time period; USE THE PERCENT OF THE ORIGINAL 100% SUBTRACTED PER PERIOD
Finding N Roots on a Calculator
Raise it to the 1/n power OR go to math and it's right there
H change (the number inside of the parenthesis in LOG FUNCTIONS)
Shifts the ASYMPTOTE along the X AXIS however many places REVERSED (if it is positive the graph is shifted toward negatives)
K Change (the number outside of the exponent after the base in EXPONENTIAL FUNCTIONS)
Shifts the ASYMPTOTE along the y axis however many places NORMALLY
K Change (the number outside of the log in LOG FUNCTIONS)
Shifts the function along the Y AXIS however many places NORMALLY
H Change (the exponent with x in EXPONENTIAL FUNCTIONS)
Shifts the function along the x axis however many places REVERSED (if it is positive the graph is shifted toward negatives)
Inverse
Switch the x and y; Symmetric about the line X=Y
b
The base; THE THING ACTUALLY BEING EXPONENTED; It is growth if b > 1 and decay if it is between 0 and 1 (or a fraction)
A(t) [Exponential Growth/Decay Formula)
The final amount after t time periods
If IaI is greater than 1 (Exponential Functions)
The graph is compressed
If IaI is in between 0 and 1 (Log Functions)
The graph is compressed vertically
If IaI is in between 0 and 1 (Exponential Functions)
The graph is expanded horizontally
If IaI is greater than 1 (Log Functions)
The graph is expanded vertically
If A is negative (Exponential/Log Functions)
The graph is reflected across the ASYMPTOTE
a [Exponential Growth/Decay Formula]
The initial amount
A (Exponential Functions)
The number before the base; It is multiplied to the base AFTER THE BASE IS EXPONENTED
r [Exponential Growth Formula]
The percent of increase per time period; USE THE PERCENT OF THE ORIGINAL 100% ADDED PER PERIOD
Logicand
The x of the log equation
When Writing An Exponential Function Whose Graph Passes 2 Points
USE FORMULA y=a b^x; A Y coordinate is the VALUE OF WHAT YOU HAVE AFTER X TIME, so you plug the FIRST Y VALUE in as the INITIAL AMOUNT (a), A VARIABLE as the BASE, the SECOND X VALUE as the EXPONENT(x), and the SECOND Y VALUE as the FINAL AMOUNT (what it all equals); JUST PLUG IT IN TO (Y=ab^x) where the variable "x" is the exponent
To Find NON CONTINUOUS Exponential Growth WITH CONSTANT INCREASE OVER TIME PERIOD THAT ISN'T COMPOUNDED
Use A(t)=a(1 + r)^t; Only used if given a fixed amount of time where the change happens like if it says "EVERY X" or "PER X"
To Find NON CONTINUOUS Exponential Decay WITH CONSTANT INCREASE OVER TIME THAT ISN'T COMPOUNDED
Use A(t)=a(1 - r)^t; Only used if given a fixed amount of time where the change happens like if it says "EVERY X" or "PER X"
Creating Like Bases
Use exponent/root rules to reduce each side to the same base and eliminate the base; Put the LIKE BASE INSIDE OF PARENTHESIS WITH THE EXPONENT VALUE THAT GIVES YOU THE SAME NUMBER AS THE ORIGINAL
If you are finding exponent bases and must use a negative exponent to make like bases
Use the negative exponent ON THE FRACTION TO GET LIKE BASES
Log Inequality Property
When converting logs with inequalities MAKE SURE THE X STAYS ON THE SAME SIDE OF THE INEQUALITY IT WAS ON BEFORE
Finding an Easy X Value (Log Graphs)
Write 4 log b logicands filled in so that you know make the answers easy, then find an x value for the function that makes the log have the easy answer you are looking for
If you have unequal bases on either side with the exponents unequal
You can create like bases and solve by multiplying the expoent of the like base exponent by the outer exponent; YOU TAKE THE TWO BASES AND FIND A NUMBER THAT CAN BE EXPONENTED BY DIFFERENT NUMBERS (Squared, Cubed) TO GET BOTH BASES (put then in parenthesis), THEN WRITE THAT NUMBER EXPONENTED BY WHATEVER MAKES IT THE BASE FOR EACH BASE, AND TO ELIMINATE THE BASES YOU MULTIPLY THE OUTSIDE EXPONENT BY THE INNER ONE
If you have the same base on both sides with different exponents
You can eliminate (take out) the base to have the EXPONENTS EQUAL to each other on either side of the equation sign
Log base x = y
base ^y = x