Introduction to Exponential Functions
What is the definition of an exponential function?
An exponential function is a mathematical function in which the variable appears in the exponent
What are the basic techniques for solving exponential equations?
The basic techniques for solving exponential equations are using properties of exponents, simplifying the equation to isolate the variable, and using graphing calculators.
The value of a smartphone depreciates at a rate of 15% per year. If the initial value is $1,000, find the smartphone's value after 3 years.
614.125
Find the Domain and Range of f(x)=5^(x)
Domain: (−∞,∞)
Range: (0,∞)
The explicit equation for a geometric sequence.
a_n=a_1(r)^(n-1)
Write the general notation for an exponential function.
The general notation for an exponential function is f(x) = a * b^x, where a is the initial value, b is the base, and x is the input variable.
How do we use a graphing calculator to solve an exponential equation?
Graph the left side and the right side of the equation on the same graph, then look for the point of intersection. The x coordinate of the point of intersection is our answer.
The value of a computer depreciates at a rate of 8% per year. If the initial value is $2,500, find the computer's value after 7 years.
1394.62
Find the Domain and Range of f(x)=6^(x) -5
Domain: (−∞,∞)
Range: (−5,∞)
The recursive equation for a geometric sequence.
Given the first term:
a_n=ra_(n-1)
How are exponential functions represented on a graph?
Exponential functions are represented on a graph as a curve that either increases (exponential growth) or decreases (exponential decay) as x increases.
Solve the exponential equation: 3^(2x) = 27.
x=3/2
The population of rabbits doubles every 6 months. If there are initially 100 rabbits, how many will there be after 2 years? Applications of Exponential Growth and Decay
1600
Find the Domain and Range of f(x)=-2^(x) +1
Domain: (−∞,∞)
Range: (−∞,1)
Write the explicit equation for the following:
4, 7, 10, 13, ....
a_n=4+(n-1)(3)
The range of an exponential function depends on these two things....
1) The horizontal asymptote, y=k. 2) If the graph is above or below the asymptote.
above -> y>k
below -> y<k
Solve the exponential equation: 2^(x + 1) = 8.
x=2
The population of a city is currently 500,000, and it is growing exponentially at a rate of 3% per year. Estimate the population after 20 years.
903,055.62
Find the Domain and Range of f(x)=-3^(x) -5
Domain: (−∞,∞)
Range: (−∞,-5)
Write the explicit equation for the following sequence:
4, -8, 16, -32, ....
a_n=4(-2)^(n-1)
Give an example of an application of exponential functions.
One example of an application of exponential functions is compound interest, where the amount of money in an investment grows exponentially over time.
(1/8)^x=32^(x-8)
x=5
Medication has a half-life of 6 hours. If a patient is given a 200 mg dose, how much will remain in their system after 24 hours?
12.5
Find the Domain and Range f(x)=-3^(x)+5
Domain: (−∞,∞)
Range: (−∞,5)
Write the recursive equation for the following:
4, -8, 16, -32, ....
a_1=4, a_n=-2a_(n-1)