Lecture Notes: Adjustments and Graphs of Logarithmic and Exponential Functions

Class Schedule Adjustments

• Apology for starting late and plan to revert to an earlier start time.
• Proposed schedule:
  • Paper 1 class at 4 PM.
  • Stats class before Maghrib.
  • Paper 3 class around 6:30-6:35 PM.
• Current issue: Delays due to S1 class starting after Maghrib.
• New plan: Move S1 to before Maghrib to start P3 earlier.

Recap of Previous Session

• Last class covered linear law and Vlogs.
• Today's focus:
  • Practice questions on linear law.
  • How to draw graphs of logarithmic (log) and exponential functions.

Linear Law

• Objective: Convert a given function to the form y = mx + c.
• Important terms:
  • "Y against X": Indicates Y on the vertical axis and X on the horizontal axis.
  • Subject of the equation is the term before "against."
• Practice question involves x^n * y = c.
  • Convert to log form: ln(x^n * y) = ln(c).
  • Use properties of logs to manipulate and simplify.
  • Solve simultaneous equations to find constants.

Example Problem

• Equation: x^n * y = c.
• Given points: (1.1, 5.2) and (3.2, 1.05).
• Convert to straight-line form to find n and c using logs.
• Gradient and intercept derived from transformed equations.

Graphs of Logarithmic and Exponential Functions

Exponential Function (y = e^x)

• Shape: Increasing curve.
• Important point: x = 0, y = 1 (e^0 = 1).
• Has a horizontal asymptote.

Logarithmic Function (y = ln(x))

• Shape: Increasing, but flattens out.
• Important point: x = 1, y = 0 (ln(1) = 0).
• Has a vertical asymptote.

Transformations

• General form for exponential: y = a * e^(bx + c) + d.
• General form for log: y = a * ln(bx + c) + d.
• Important points for graphs:
  • Exponential: Power equals zero for pivotal point.
  • Logarithmic: Argument equals one for pivotal point.

Practical Application

• Example: Draw graph for y = 2e^(x-2).
• Method:
  • Find pivotal point (x = 2, y = 2).
  • Plot additional points around pivotal point.
  • Consider asymptote and possible transformations.
• Example: Draw graph for y = -2ln(x + 4).
  • Find pivotal point (x = -3, y = 0).
  • Plot points and asymptote.

Conclusion

• Key takeaway: Identifying pivotal points and asymptotes are crucial for graphing.
• Graph shapes have limited variations based on transformations.
• Practice using table functions on calculators for efficiency.

Additional Notes

• Class questions and clarifications were addressed.
• Emphasis on using important points for efficient graphing.
• Reminder about iteration topic and its relevance to graphing.