Lecture Notes: Graph Sketching and Analysis

Introduction

• Lecture by Mr. Hassan on graph sketching.
• Focus on a question from January 2024 International A-Level Ed Excel Pure Mathematics P1 paper.

Graph Sketching Basics

• Equation: y = 4/(x - K), where K is a positive constant.
• Objective: Sketch the graph, showing points where it cuts axes and vertical asymptote.

Identifying Graph Types

• Linear Graphs: y = mx + C
  • Straight line, X has power 1.
• Quadratic Graphs: y = ax^2 + bx + C
  • Parabola shape, highest power is x^2.
• Cubic Graphs: y = ax^3 + bx^2 + cx + d
  • Up-down-up or down-up-down shape.
• Reciprocal Graphs: y = something/x
  • Discontinuity at certain values, X in the denominator.

Analyzing the Given Curve

• Equation Analysis: y = 4/(x - K) is a reciprocal curve.
• Vertical Asymptote: X = K
  • Occurs where the denominator is zero.
• Horizontal Asymptote: y = 0
  • Occurs as y cannot be zero.

Sketching the Curve

• Axes Setup: Y-axis and X-axis.
• Vertical Asymptote at X = K: Positive side, due to positive constant K.
• Crossing Y-axis: At point (0, -4/K).
• Curve Shape: Reciprocal shape, asymptotes guide the curve.

Question Part B: Line and Curve Intersection

• Line Equation: y = 9 - x
  • Does not cross or touch the curve.
• Finding Range of K: Ensure no solutions to simultaneous equations.

Solving Simultaneous Equations

• Substitute Line Equation: 9 - x = 4/(x - K)
• Expanding and Simplifying: Leads to quadratic inequality.
• Discriminant Analysis: b^2 - 4ac < 0 for no solution (no intersection).

Solving for K

• Quadratic Form: a = 1, b = -(9 + K), c = 9K + 4
• Critical Values: Solve K^2 - 18K + 65 = 0
  • Factorize and identify critical values: K = 5 and K = 13.
• Inequality Solution: K is between 5 and 13 for no intersection.

Conclusion

• Understanding Intersections: Solve equations simultaneously.
• Using Discriminant: Key for finding values of K ensuring no line-curve intersection.
• Graph Sketching Skills: Important for analyzing intersections and asymptotes.

Additional Resources

• Further questions and topics on graph sketching available in playlists.