Linear Equations

• A linear equation is an equation in which the highest power of the variable(s) is 1.
• Standard form: Ax + By = C, where A, B, and C are integers, and A and B are not both zero.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
• Graphing linear equations:
  • The graph of a linear equation is a straight line.
  • The slope (m) determines the steepness and direction of the line.
  • The y-intercept (b) determines the point at which the line crosses the y-axis.

Graphing Functions

• The graph of a function is a visual representation of the relationship between the input (x) and output (y) values.
• The graph can be used to:
  • Identify the domain and range of the function.
  • Determine the x-intercepts (roots) of the function.
  • Identify the y-intercept of the function.
  • Analyze the behavior of the function (increasing, decreasing, maxima, minima).

Exponential Growth

• Exponential growth occurs when a quantity increases by a fixed percentage at regular time intervals.
• Exponential growth formula: A = P(1 + r/n)^(nt), where:
  • A is the amount of money accumulated after n years, including interest.
  • P is the principal amount (initial investment).
  • r is the annual interest rate (in decimal form).
  • n is the number of times that interest is compounded per year.
  • t is the time the money is invested or borrowed for, in years.
• Characteristics of exponential growth:
  • Rapid acceleration in growth rate.
  • Eventual surpassing of linear growth.

Quadratic Functions

• A quadratic function is a polynomial function of degree two, meaning the highest power of the variable(s) is two.
• Standard form: f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0.
• Graphing quadratic functions:
  • The graph of a quadratic function is a parabola that opens upward or downward.
  • The vertex of the parabola is the minimum or maximum point of the function.
  • The axis of symmetry is the vertical line that passes through the vertex.
• Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, which solves quadratic equations of the form ax^2 + bx + c = 0.

Function Operations

• Function operations allow us to combine functions in different ways to create new functions.
• Types of function operations:
  • Function addition: (f + g)(x) = f(x) + g(x)
  • Function subtraction: (f - g)(x) = f(x) - g(x)
  • Function multiplication: (f × g)(x) = f(x) × g(x)
  • Function division: (f ÷ g)(x) = f(x) ÷ g(x), as long as g(x) ≠ 0
• Composition of functions: (f ∘ g)(x) = f(g(x)), where the output of g becomes the input for f.