Exploring Quadratic Relations: Graphing

Mini Golf Example (Exercise 6.1 on Page 322)

• Quadratic Equation: ( y = -2.5x^2 + 6x )
  • Y: Represents distance in meters.
  • X: Represents time in seconds.
• Graphing the Quadratic Relation:
  • X-axis: Time (seconds)
  • Y-axis: Distance from starting point (meters)
  • Graph presents a parabola (curve that opens or closes based on its equation).
  • Use a graphing calculator by entering the quadratic equation and setting an appropriate window:
    • Window: X from 0 to 3, Y from 0 to 5.

Characteristics of Quadratic Relations

• Shape: Parabola
• Standard Equation: ( y = ax^2 + bx + c )
  • a, b, c: Real numbers.
  • a: Affects shape (narrower or wider) and direction (opens up if positive, down if negative).
  • b: Moves the graph horizontally.
  • c: Vertical shift (up if positive, down if negative).

Graph Analysis

• Effect of 'a':
  • Increasing 'a' makes the graph narrower or steeper.
  • Negative 'a' flips the graph upside down.
• Effect of 'b':
  • Moves the vertex around horizontally.
• Effect of 'c':
  • Directly correlates with the Y-intercept.

Common Characteristics of Parabolas

• Symmetry: About a vertical line through the vertex.
• Vertex: The highest or lowest point depending on the direction of opening.
• Axis of Symmetry: Line running through the vertex.
• Functions: Each quadratic relation represents a function.
• Degree: Quadratic equations are degree 2.

Predicting Parameter Values

• Effect of Parameters:
  • a: Affects the steepness and direction of opening.
  • b: Horizontal movement of the graph without altering its shape.
  • c: Directly identifies as the Y-intercept.

Practical Usage

• Use a graphing calculator for accurate plotting.
• Adjust parameters to understand their influence on the quadratic curve's characteristics.

Summary

• A quadratic relation forms a parabola with features defined by parameters ( a, b, \text{and} c ).
• These parameters influence the graph's shape, direction, position, and intercepts.
• Understanding these effects allows for prediction and manipulation of the graph for diverse quadratic problems.