Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Finite Differences Investigations Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

Finite Differences First Difference: Values that are calculated by subtracting consecutive y – values in a table of values that has a constant difference between the x – values. Finite Differences: Differences between the y – values in a table of values in which the x – values increase by the same amount.  Relation: A description of how two variables are connected. Linear Relation: A relation whose graph is a straight line and whose first differences are constant. Non-linear Relation: A relation whose graph is a not straight line and whose first differences are not constant. Quadratic Relation: A relation whose graph is a parabolic shape and whose second differences are constant. Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Patterns The first four diagrams in a pattern show number of apples are contained in each basket.  a) Complete the table according to the given diagram. Guess the number of apples in the baskets 5 & 6. Basket # Number of apples First Differences 1 2 3 4 5 6 5 8 11 14 8 – 5 = 3 11 – 8 = 3 14 – 11 = 3 17 20 3 Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Patterns - Continued Use finite differences to determine whether the function is a linear or quadratic. Basket # Number of apples First Differences 1 5 2 8 3 11 4 14 17 6 20 Linear Determine an equation of the relation. How many apples will there be in the 10th basket? the 35th basket? There will be 32 apples in the 10th basket. There will be 107 apples in the 35th basket. Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 1: Patterns - Continued Which basket contains 128 apples? (Assume the basket can hold infinite apples!! ) Recall: The 42nd basket contains 128 apples. Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 2: Patterns The first four diagrams in a pattern show number of worms grow during the first 4 weeks  a) Complete the table according to the given diagram. Guess the number of worms in the weeks 5 & 6. Week # Number of Worms First Differences Second Differences 1 2 3 4 5 6 2 5 10 17 5 – 2 = 3 10 – 5 = 5 5 – 3 = 2 17 – 10 = 7 7 – 5 = 2 26 37 9 11 2 Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 2: Patterns - Continued Use finite differences to determine whether the function is a linear or quadratic. Week # Number of Worms First Differences Second Differences 1 2 5 3 10 4 17 7 26 9 6 37 11 Non- Linear  Quadratic Determine an equation of the relation. (By observation) How many worms will there be in the 9th week? the 20th week? There will be 82 worms in the 9th week. There will be 401 worms in the 20th week. Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 2: Patterns - Continued Which week will have 1025 worms? Week # Number of Worms First Differences Second Differences 1 2 5 3 10 4 17 7 26 9 6 37 11 Week 32 will have 1025 worms. f)  Use the table to create a quadratic relation. Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Example 3: Patterns Use the finite differences to determine the relation is linear or quadratic or neither, then determine the relation. x y First Differences Second Differences 1 -2 2 -11 3 -26 4 -47 -3 -9 -6 -15 -21 Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Finite Differences Investigations Text: P. 254 #2, 4, 13, 14 Check the website for updates Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Finite Differences Investigations © 2017 E. Choi – MPM2D - All Rights Reserved