뒤로Precalculus Study Guide: Linear, Quadratic, Exponential, and Logarithmic Functions
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Linear, Quadratic, Exponential, and Logarithmic Functions
Linear Functions
Linear functions are algebraic expressions of the form f(x) = mx + b, where m is the slope and b is the y-intercept. They model relationships with constant rates of change.
Definition: A function whose graph is a straight line.
General Form:
Example: If and 3 is added to y for each , then , , so .
Application: Used to model constant speed, cost, or other linear relationships.
Exponential Functions
Exponential functions have the form f(x) = ab^{x}, where a is the initial value and b is the base or growth factor. They model growth or decay processes.
Definition: A function in which the variable appears in the exponent.
General Form:
Example: If and y is multiplied by 3 for each , then , , so .
Application: Population growth, radioactive decay, compound interest.
Quadratic Functions
Quadratic functions are polynomials of degree 2, typically written as f(x) = ax^2 + bx + c. Their graphs are parabolas.
Definition: A function where the highest power of x is 2.
General Form:
Example: If zeros are at and , and , use factored form: . Plug to solve for :
So,
Application: Projectile motion, area problems.
Secant Line
The secant line to a curve is a straight line that passes through two points on the curve. For on the interval [1, 100]:
Definition: A line connecting two points and .
Formula for slope:
Example: For , , , (common log):
Slope:
Equation:
Population Growth Model
Exponential growth models describe populations that grow at a constant percentage rate per time period.
General Form:
Where: is the initial population, is the growth rate (as a decimal), is time.
Example: For 5% growth, :
Doubling Time: Set and solve for :
Graph Analysis and Properties
Linear Graphs
Example:
Slope: (decreasing function)
Y-intercept:
Exponential Graphs
General Form: , ,
Increasing/Decreasing: If , increasing; if , decreasing.
Concavity: Exponential functions are always concave up.
End Behavior: As , (if ); as , .
Logarithmic Graphs
General Form: ,
Increasing/Decreasing: Increasing for
Concavity: Concave down
Key Points: and
Quadratic Graphs
General Form:
Concavity: Up if , down if
Vertex:
Example: is concave down, vertex at
Logarithms: Properties and Calculations
Logarithm Basics
Definition: is the exponent to which must be raised to get .
Common Logarithm: Base 10,
Natural Logarithm: Base ,
Evaluating Logarithms
(since )
(since )
is undefined (logarithm of a negative number is not real)
(since )
(since )
Logarithm Properties
Product Rule:
Quotient Rule:
Power Rule:
Change of Base:
Simplifying Logarithmic and Exponential Expressions
Example:
Example: (since )
Solving Exponential and Logarithmic Equations
Exponential Equations:
Logarithmic Equations:
Mixed Equations:
Natural Logarithm Equation:
Summary Table: Function Types and Properties
Function Type | General Form | Key Properties | Example |
|---|---|---|---|
Linear | Constant rate of change, straight line | ||
Quadratic | Parabola, vertex, zeros | ||
Exponential | Growth/decay, constant percent change | ||
Logarithmic | Inverse of exponential, slow growth |
Additional info:
Some context and explanations have been expanded for clarity and completeness.
Examples and formulas are provided for each function type to aid understanding.
