Graphs: Linear, Quadratic, and Trigonometric

Linear Graphs

• Definition: A graph representing a linear equation (y = mx + b).
• Slope (m): Indicates the steepness and direction of the line.
  • Positive slope: upward line.
  • Negative slope: downward line.
• Y-intercept (b): Point where the graph crosses the y-axis.
• Key Characteristics:
  • Straight line.
  • Continuous with no curves.
  • Domain and range: all real numbers.

Quadratic Graphs

• Definition: A graph of a quadratic function (y = ax^2 + bx + c).
• Shape: Parabola (opens upward if a > 0, downward if a < 0).
• Vertex: The highest or lowest point of the parabola.
• Axis of Symmetry: A vertical line through the vertex.
• X-intercepts: Points where the graph crosses the x-axis (found using the quadratic formula).
• Y-intercept: Point where x = 0 (y = c).

Trigonometric Graphs

• Functions: Sine (sin), Cosine (cos), Tangent (tan).
• Periodicity: Trig functions are periodic, with specific periods.
  • Sin and Cos: Period of 2π.
  • Tan: Period of π.
• Key Characteristics:
  • Amplitude: Maximum vertical distance from the midline (for sin and cos).
  • Phase Shift: Horizontal shift of the graph.
• Key Points:
  • Sin(0) = 0, Cos(0) = 1, Tan(0) = 0.
  • Sin(π/2) = 1, Cos(π/2) = 0, Tan(π/4) = 1.

Exponential Graphs

• Function: Generally in the form y = a * b^x (a > 0, b > 1).
• Growth: Rapid increase (if b > 1) or decrease (if 0 < b < 1).
• Y-intercept: Always at (0, a).
• Asymptote: The horizontal line y = 0 (x-axis) acts as a horizontal asymptote.
• Key Characteristics:
  • Domain: All real numbers.
  • Range: y > 0.

Logarithmic Graphs

• Function: Inverse of exponential functions, typically y = log_b(x).
• Characteristics:
  • Vertical asymptote at x = 0.
  • Passes through (1, 0) since log_b(1) = 0.
• Growth: Slowly increases; grows without bound but never touches the y-axis.
• Domain and Range:
  • Domain: x > 0.
  • Range: All real numbers.

Linear Graphs

• Represented by a linear equation: y = mx + b
• Slope (m) determines steepness and direction of the line:
  • Positive slope: upward line
  • Negative slope: downward line
• Y-intercept (b) is where the graph crosses the y-axis
• Always a straight line, continuous with no curves
• Domain and range: all real numbers

Quadratic Graphs

• Represented by a quadratic function: y = ax^2 + bx + c
• Shaped like a parabola
  • Opens upward if a is positive
  • Opens downward if a is negative
• Vertex: highest or lowest point of the parabola
• Axis of Symmetry: vertical line passing through the vertex
• X-intercepts: points where the graph crosses the x-axis, found using the quadratic formula
• Y-intercept: point where x = 0 (y = c)

Trigonometric Graphs

• Functions: Sine (sin), Cosine (cos), Tangent (tan)
• Periodic: repeat in cycles
  • Sin and Cos: period of 2π
  • Tan: period of π
• Key aspects:
  • Amplitude: maximum vertical distance from the midline (for sin and cos)
  • Phase Shift: horizontal shift of the graph
  • Key Points:
    • Sin(0) = 0, Cos(0) = 1, Tan(0) = 0
    • Sin(π/2) = 1, Cos(π/2) = 0, Tan(π/4) = 1

Exponential Graphs

• Generally in the form: y = a * b^x (where a > 0 and b > 1)
• Exhibits rapid growth if b > 1, rapid decrease if 0 < b < 1
• Y-intercept always at (0, a)
• Horizontal asymptote at y = 0 (x-axis)
• Domain: all real numbers
• Range: y > 0

Logarithmic Graphs

• Function: Inverse of exponential functions, typically y = log_b(x)
• Characteristics:
  • Vertical asymptote at x = 0
  • Passes through (1, 0) since log_b(1) = 0
  • Slowly increases; grows without bound but never touches the y-axis
• Domain: x > 0
• Range: all real numbers