Math Apr 7

  • Pizza and Health

    • The question posed: Is pizza healthy?

    • Enthusiastic response in affirmation.

  • Using Desmos for Calculations

    • Suggestion to use Desmos for quick calculations.

    • Focus on solving for y value; potential elimination method.

  • Measurement Systems

    • Discussion on measurement systems (kilometers, meters, centimeters) using base 10.

    • Example of U.S. measurements (inches, yards, miles) that deviate from base 10.

    • Common log is in base 10.

  • Exponential Functions

    • Transition to working with exponents and logarithms.

    • Importance of recognizing that base 10 is assumed if not specified.

    • Example:

    • 10y=100010^{y} = 1000

    • Interpreted as 10y=10310^{y} = 10^{3}

    • Move into denominators necessitates negative exponents, e.g., 10−310^{-3}.

    • Both expressions maintain the same base 10.

  • Using Calculators for Logarithms

    • Encouraged to verify calculations with a calculator for log values.

    • Example of calculating {log}(15) leavingleaving 10^{y} = 15

    • Solution round-up noted to be 1.176.

    • Importance of accurate rounding for higher math levels.

  • Introduction to Natural Logarithms

    • Introduction of natural logarithm function: f(x)=ln(x)f(x) = {ln}(x).

    • Explanation of the constant ee, which is approximately 2.718.

    • Significance of understanding logarithms for future studies (college algebra & beyond).

  • Function Definition

    • Reminder of function definition: Each input has exactly one output.

    • Importance of parentheses in defining function behavior.

  • Logarithms as Inverses of Exponentials

    • Discussion of switching X and Y values when using logarithmic transformations.

    • Explaining the process to calculate outputs from inverting exponential functions.

    • Understanding of how to navigate transformations in functions.

  • Graphing Logarithmic Functions

    • Process of creating tables for parent functions and their transformations.

    • Involving negative values and transformations affecting both horizontal and vertical shifts.

    • Recognition of asymptotes (vertical and horizontal) in graphs, specifically moving from horizontal asymptotes of exponentials to vertical asymptotes in logarithmic contexts.

    • Explanation of inverses flipping domain and range characteristics.

  • End Behavior of Functions

    • Distinction in end behavior of functions when reaching vertical asymptotes evaluated.

    • Noting that x values in logarithmic functions won't approach infinity on both sides like in previous functions.

    • Use of vertical asymptotes to assess left-side behavior in graphs.

  • Conclusion and Summation

    • Reference to completion of all needed tables and transformations forming base for future problems.

    • Emphasis on transformations affecting outcomes when graphing functions.