Nonlinear functions are the secret sauce of math. They're not just straight lines – they curve, bend, and twist in ways that can model real-world stuff like population growth or falling objects. Understanding these functions is key to acing the SAT Math section.

From exponential growth to quadratic curves and rational functions, nonlinear math is all about patterns and relationships. Knowing how to spot these patterns, graph them, and solve related equations will give you a major edge on test day. Let's dive into the world of curves and asymptotes!

Characteristics of Exponential Functions

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Basic Structure and Properties

• Exponential functions have the general form $f(x) = a \cdot b^x$, where a is the initial value and b is the base (growth factor)
• Graph of an exponential function never crosses the x-axis
• Y-intercept of an exponential function always equals the initial value (a)
• Exponential functions exhibit a constant percent rate of change over equal intervals

Growth and Decay Behavior

• When $b > 1$, the function represents exponential growth (increasing)
• When $0 < b < 1$, the function represents exponential decay (decreasing)
• Growth functions increase rapidly, while decay functions approach zero asymptotically

Exponential Growth and Decay

Modeling Real-World Scenarios

• Exponential growth and decay modeled using the equation $A(t) = A_0 \cdot (1 + r)^t$ or $A(t) = A_0 \cdot e^{rt}$
  • $A_0$ represents the initial amount
  • r denotes the growth or decay rate
  • t signifies the time
• Doubling time (growth) or half-life (decay) calculated using $t = \frac{\ln(2)}{\ln(1 + r)}$
• Applications include population growth, radioactive decay, and compound interest

Financial Applications

• Continuously compounded interest calculated using $A(t) = P \cdot e^{rt}$
  • P represents the principal amount
  • r denotes the annual interest rate
  • t signifies the number of years
• Used in financial planning, investment analysis, and loan calculations (mortgages, student loans)

Key Features of Quadratic Graphs

Parabola Characteristics

• Quadratic functions have the general form $f(x) = ax^2 + bx + c$, where $a \neq 0$
• Graph of a quadratic function forms a parabola
• Parabola opens upward when $a > 0$ (minimum point)
• Parabola opens downward when $a < 0$ (maximum point)
• Vertex represents the minimum or maximum point with coordinates $(h, k)$
  • $h = -\frac{b}{2a}$ and $k = f(h)$

Important Points and Lines

• Axis of symmetry passes through the vertex, given by $x = -\frac{b}{2a}$
• Y-intercept occurs where the parabola intersects the y-axis $(0, c)$
• X-intercepts (roots or zeros) are points where the parabola intersects the x-axis
  • Can have 0, 1, or 2 x-intercepts depending on the discriminant

Solving Quadratic Equations

Solution Methods

• Quadratic equations solved by factoring, completing the square, using the quadratic formula, or graphing
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Used when a, b, and c are known in $ax^2 + bx + c = 0$
• Factoring method effective for equations with integer solutions
• Completing the square useful for finding the vertex form of a quadratic function

Discriminant Analysis

• Discriminant (Δ) determines the number and type of solutions: $\Delta = b^2 - 4ac$
• If $\Delta > 0$, two distinct real solutions exist (parabola crosses x-axis twice)
• If $\Delta = 0$, one repeated real solution occurs (parabola touches x-axis once)
• If $\Delta < 0$, two complex solutions result (parabola doesn't intersect x-axis)

Properties of Polynomial Functions

General Characteristics

• Polynomial functions have the form $f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$
  • $a_0, a_1, ..., a_n$ are constants and n is a non-negative integer
• Degree of a polynomial is the highest power of the variable (n)
• Leading coefficient is the coefficient of the term with the highest degree ($a_n$)
• Polynomial functions are continuous with no asymptotes
• Graphs can have multiple turning points and intersections with x-axis

Behavior and Applications

• End behavior determined by the degree and sign of the leading coefficient
• Used to model various phenomena (projectile motion, economic trends)
• Higher-degree polynomials can approximate complex curves and data sets

End Behavior and Zeros of Polynomials

End Behavior Patterns

• Even-degree polynomials:
  • Positive leading coefficient: graph rises on both ends
  • Negative leading coefficient: graph falls on both ends
• Odd-degree polynomials:
  • Positive leading coefficient: graph falls on left, rises on right
  • Negative leading coefficient: graph rises on left, falls on right

Zeros and Their Properties

• Zeros (roots) are x-values where $f(x) = 0$
• Multiplicity of a zero affects graph behavior near x-intercept:
  • Odd multiplicity: graph crosses x-axis
  • Even multiplicity: graph touches but doesn't cross x-axis
• Number of zeros ≤ degree of polynomial
• Complex zeros always occur in conjugate pairs

Rational Functions and Graphs

Function Characteristics

• Rational functions have the form $f(x) = \frac{P(x)}{Q(x)}$, where P(x) and Q(x) are polynomials and $Q(x) \neq 0$
• Domain excludes x-values that make denominator zero
• Graphs can have vertical asymptotes, horizontal asymptotes, and holes

Asymptote Analysis

• Vertical asymptotes occur where denominator equals zero
• Horizontal asymptotes:
  • Degree of numerator < degree of denominator: y = 0
  • Degree of numerator = degree of denominator: y = ratio of leading coefficients
• Oblique (slant) asymptotes occur when degree of numerator is one more than degree of denominator
• Asymptotes help predict function behavior for large x values

Solving Rational Equations

Solution Process

• Identify x-values making denominators zero (not solutions)
• Multiply both sides by least common denominator (LCD) to clear fractions
• Solve resulting polynomial equation
• Check potential solutions in original equation
• Discard solutions making denominators zero

Solution Interpretation

• Consider context of problem when interpreting solutions
• Ensure solutions fall within domain of original function
• Analyze reasonableness of solutions in real-world scenarios (negative time, impossible quantities)
• Graph solutions to visualize their relationship to the function