Exponential functions model growth and decay, with bases determining their behavior. They're crucial in finance, population studies, and physics. These functions increase or decrease rapidly, never touching zero, and have wide-ranging applications in real-world scenarios.
Mastering exponentials involves understanding their properties, graphing techniques, and solving related equations. This knowledge forms the foundation for more advanced mathematical concepts and practical problem-solving in various fields, from economics to engineering.
Exponential Functions
Exponential functions with various bases
defined as f(x)=bx, where b is the base and x is the exponent
Base b must be positive and not equal to 1 (b>0 and b=1)
Function decreases as x increases when base is between 0 and 1 (0<b<1)
Function increases as x increases when base is greater than 1 (b>1)
Natural exponential function f(x)=ex uses mathematical constant e (approximately 2.71828) as base
Unique property of being its own derivative makes it useful in calculus and mathematical modeling
Solving equations with exponential functions involves isolating exponential term on one side
Take logarithm of both sides using appropriate base to solve for variable (logarithms)
Derivation of exponential equations
equation A(t)=A0⋅bt models increase over time t from initial amount A0
Doubling time td=ln(b)ln(2) calculates time needed for quantity to double
Half-life th=ln(b)ln(0.5) determines time required for quantity to reduce by half
equation A(t)=A0⋅b−t represents decrease over time from initial amount
Given two points (x1,y1) and (x2,y2), exponential function equation can be found
Substitute points into y1=A0⋅bx1 and y2=A0⋅bx2
Solve system of equations for A0 and b
Substitute calculated values into general form f(x)=A0⋅bx
Applications of exponential growth
formula A(t)=P(1+nr)nt calculates growth of principal P at annual interest rate r compounded n times per year over time t
Continuously compounded interest uses formula A(t)=Pert
Population growth modeled by P(t)=P0⋅ekt with initial population P0, growth rate k, and time t
Radioactive decay represented by N(t)=N0⋅e−λt with initial amount N0, decay constant λ, and time t
Graphing and interpreting exponentials
Domain includes all real numbers (−∞,∞)
Range consists of all positive real numbers (0,∞)
Horizontal asymptote at y=0 as x approaches negative infinity (asymptotic behavior)
y-intercept at (0,1) for all exponential functions
Function increases when b>1 and decreases when 0<b<1
No x-intercept exists for exponential functions
Graph is always concave up regardless of base value
Interpreting exponential graphs:
y-value increases or decreases by factor of b for each unit increase in x
Larger base b results in more rapid growth or decay of function
Rate of change of exponential functions is proportional to the function itself
Inverse Functions and Logarithms
Exponential functions have inverse functions called logarithms
Logarithmic functions are defined as the inverse of exponential functions
The domain and range of exponential functions and their logarithmic inverses are reversed
Key Terms to Review (7)
Annual percentage rate (APR): Annual Percentage Rate (APR) is the annual rate charged for borrowing or earned through an investment, expressed as a percentage. It represents the yearly cost of funds over the term of a loan or income earned on an investment.
Compound interest: Compound interest is the interest calculated on the initial principal, which also includes all accumulated interest from previous periods. It can be expressed by the formula $A = P(1 + \frac{r}{n})^{nt}$ where A is the amount of money accumulated after n years, including interest.
Exponential decay: Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. Mathematically, it is expressed as $N(t) = N_0 e^{-kt}$, where $N_0$ is the initial quantity, $k$ is the decay constant, and $t$ is time.
Exponential function: An exponential function is a mathematical function in the form $f(x) = a \cdot b^x$, where $a$ is a constant, $b$ is the base greater than 0 and not equal to 1, and $x$ is the exponent. These functions model growth or decay processes.
Exponential growth: Exponential growth describes a process where the quantity increases at a rate proportional to its current value. This results in the quantity growing faster and faster over time.
Linear growth: Linear growth is a pattern of increase at a constant rate over equal intervals of time. It can be represented by the equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Nominal rate: The nominal rate is the interest rate before adjustments for inflation or other factors. It is often used in exponential growth calculations where interest compounds over time.