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Understanding Rate of Change: Linear vs. Exponential Functions

This week’s lesson series focuses on the concept of Rate of Change (ROC). We'll explore how to determine the domain and range of functions, analyze where graphs are positive, and apply these concepts to real-world problems, such as the cost of renting a canoe. We'll emphasize the differences between linear and exponential rates of change through discussions, quizzes, and problem-solving activities. Prepare for a test next Tuesday as we solidify our understanding of ROC through graphs, tables, and equations.

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Understanding Rate of Change: Linear vs. Exponential Functions

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  1. Tuesday, January 21st Warm Up What is the domain and Range? Where is the graph positive?

  2. EOCT Week 1 Winners

  3. Week 2 EOCT-#1 To rent a canoe, the cost is $3 for the oars and life preserver, plus $5 an hour for the canoe. Which graph models the cost of renting a canoe?

  4. Quiz Scores/Answers

  5. A Look Ahead Tuesday: Rate of Change-Linear Wednesday: Rate of Change-Exponential Thursday: Rate of Change word Problems Friday: Rate of Change word Problems *TEST next Tuesday!

  6. What words do you think of when you think of rate of change?!?

  7. Finding Rate of Change • Tables • Graphs • Equations • Word Problems

  8. Group Discussion How would you describe the DIFFERENCE between the rate of change between: Exponential & Linear Functions

  9. Linear Exponential

  10. ROC Linear Exponential Rate of Change=constant change by MULTIPLICATION Rate of change depends on the interval you are looking at! • Rate of change=CONSTANT CHANGE ADDING OR SUBTRACTING • At any interval the rate of change will be the same!

  11. LINEAR Key Concepts • Slope=rate of change • Positive ROC=Increasing • Negative ROC=Decreasing • ROC of ZERO=Horizontal line • ROC of UNDEFINED=vertical Line

  12. Determine if ROC is Postive, Negative, zero, undefined-Graphs N P U 0

  13. Determine if ROC is Postive, Negative, zero, undefined-Equations 0 • Y=8 • Y=3x + 7 • X= 8 • Y=-3x + 7 P U N

  14. Determine if ROC is Postive, Negative, zero, undefined-Tables P 0 N U

  15. Important: Must solve for y!

  16. Find the Rate of Change • Y = 2x – 10 • Y = -5x + 4 • 2x = 4y + 16 • 2y + 6x = 24

  17. Remember: Two negatives=Positive - -=+

  18. ALWAYS

  19. 1. 2.

  20. 3. 4.

  21. Finding ROC from Graphs

  22. I. Rise and Run Rise= y axis Run= x axis

  23. What goes first? You need to rise up before you run

  24. Run –9 • Rise 3 Rise –3 • Run 9 Finding Slope with a Graph Find the slope of the line • Begin at one point and count vertically to fine the rise. • Up=Positive • Down=Negative (–6, 5) • Then count horizontally to the second point to find the run. • Right=Positive • Left= Negative (3, 2) It does not matter which point you start with. The slope is the same.

  25. Run –5 Rise –2 Rise 2 Run 5 Begin at one point and count vertically to find rise. Remember: up=____ down=___ Then count horizontally to the second point to find the run Remember: right=____left=_____ • •

  26. You Try!

  27. SUMMARY • What type of function has a ROC that can be described as a constant change adding or subtracting • How do you find the ROC in a linear equation? • How do you find the ROC in a linear table? • How do you find the ROC on a linear graph?

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