Linear functions form the foundation of calculus, with their constant rate of change. Slope, the key feature, measures steepness and direction. Understanding linear functions is crucial for grasping more complex relationships in mathematics.

Polynomial functions expand on linear concepts, introducing higher degrees and varied behaviors. These functions exhibit distinct characteristics based on their degree and coefficients. Mastering polynomials prepares you for analyzing more intricate mathematical relationships and real-world applications.

Linear Functions

Slope of linear functions

• Measures the rate of change or steepness of a linear function
• Calculated as the change in y-values divided by the change in x-values $\frac{\Delta y}{\Delta x}$
• Remains constant for a linear function
• Positive slope indicates an increasing function (line rises from left to right)
• Negative slope indicates a decreasing function (line falls from left to right)
• Zero slope represents a horizontal line (no change in y-values)
• Steeper lines have larger absolute values of slope

Polynomial Functions

Polynomial function characteristics

• Consist of terms with non-negative integer exponents
• Degree is the highest exponent of the variable
• Linear functions are first-degree polynomials ($ax + b$)
• Quadratic functions are second-degree polynomials ($ax^2 + bx + c$)
• Odd-degree polynomials:
  • Have at least one real root (x-intercept)
  • Exhibit opposite end behavior ($x \to -\infty$, $f(x) \to \pm\infty$ and $x \to \infty$, $f(x) \to \mp\infty$)
• Even-degree polynomials:
  • May not have any real roots
  • Exhibit the same end behavior ($x \to \pm\infty$, $f(x) \to \infty$ for positive leading coefficient or $f(x) \to -\infty$ for negative leading coefficient)

Quadratic equations and roots

• In the form $ax^2 + bx + c = 0$, where $a \neq 0$
• Roots are x-values where the function equals zero
• Roots found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• Roots are x-coordinates of points where the graph crosses the x-axis
• Discriminant ($b^2 - 4ac$) determines the nature of the roots:
  • Positive discriminant indicates two distinct real roots
  • Zero discriminant indicates one repeated real root
  • Negative discriminant indicates two complex conjugate roots

Other Functions

Types of algebraic functions

• Rational functions are ratios of two polynomials
  • May have vertical asymptotes where the denominator equals zero
  • May have horizontal asymptotes as $x \to \pm\infty$
• Power functions are of the form $f(x) = x^a$, where $a$ is constant
  • Positive $a$ results in an increasing function
  • Negative $a$ results in a decreasing function
  • Even $a$ results in a function symmetric about the y-axis
  • Odd $a$ results in a function symmetric about the origin
• Root functions are of the form $f(x) = \sqrt[n]{x}$, where $n$ is a positive integer
  • Domain is all non-negative real numbers
  • Graph is increasing and concave down

Algebraic vs transcendental functions

• Algebraic functions constructed using finite algebraic operations (addition, subtraction, multiplication, division, roots)
  • Include polynomial, rational, power, and root functions
• Transcendental functions are not algebraic
  • Include exponential, logarithmic, and trigonometric functions
  • Often have unique characteristics (asymptotes, periodicity)

Piecewise function graphing

• Defined by different expressions for different domain intervals
• To graph:
  1. Identify the domain intervals for each piece
  2. Graph each piece on its respective interval
  3. Use open or closed circles to indicate endpoint inclusion or exclusion

Function graph transformations

• Vertical shifts:
  • $f(x) + k$ shifts the graph up by $k$ units
  • $f(x) - k$ shifts the graph down by $k$ units
• Horizontal shifts:
  • $f(x - h)$ shifts the graph right by $h$ units
  • $f(x + h)$ shifts the graph left by $h$ units
• Vertical stretches and compressions:
  • $af(x)$ stretches the graph vertically by $|a|$ if $|a| > 1$
  • $af(x)$ compresses the graph vertically by $|a|$ if $0 < |a| < 1$
• Horizontal stretches and compressions:
  • $f(bx)$ compresses the graph horizontally by $|b|$ if $|b| > 1$
  • $f(bx)$ stretches the graph horizontally by $|b|$ if $0 < |b| < 1$
• Reflections:
  • $-f(x)$ reflects the graph over the x-axis
  • $f(-x)$ reflects the graph over the y-axis

Function Properties and Operations

Domain and range

• Domain: Set of all possible input values (x-values) for which the function is defined
• Range: Set of all possible output values (y-values) that result from the function

Function composition and inverse

• Function composition (f ∘ g)(x) combines two functions by applying one function to the output of another
• Inverse function f⁻¹(x) "undoes" the original function, swapping input and output values
  • Not all functions have inverses

Continuity and limits

• A function is continuous at a point if there are no gaps, jumps, or holes in its graph
• Discontinuity occurs when a function is not continuous at a point
• Limit describes the behavior of a function as the input approaches a specific value
  • Helps analyze function behavior near points of discontinuity