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

Test your understanding of different types of graphs including linear, quadratic, and trigonometric. The quiz covers definitions, key characteristics, and properties of each graph type. Challenge yourself to apply these concepts to various scenarios.