Report

4.3 Patterns and Non-Linear Functions: Nonlinear: Is a function whose graph is not a line or part of a line. Family of Functions: is a group of functions with common characteristics. Parent Function: is the simplest function of a family of functions. GOAL: One of the ways to classify a function is to see a visual representation of it. A graph would show the function’s characteristics and would provide us with info to classify it as: 1. Linear : Graph is nonvertical line or part of a nonveritcal line 2. Nonlinear Function: Graph whose part is not a line or part of a line. 1. Linear : Graph is nonvertical line or part of a nonveritcal line Insert Parent Graphs Page 246 Linear Parent Functions: y = mx + b y=n y = -mx + b x=n 2. Nonlinear Function: Graph whose part is not a line or part of a line. Nonlinear Parent Functions: y = x2 y = x3 y=|x| Nonlinear Parent Functions: y= y = ax if a> 1 y = ax 0<a<1 LINEAR OR NONLINEAR?: The area A, of a pizza is a function of its radius r, in inches. The cost C, in dollars, of the sauce for a pizza is a function of the weight w, in ounces, of sauce used. Graph this functions and classify as linear on nonlinear. Sauce Cost Pizza Area Radius (in.) , r Area (in2), A Weight (oz), w Cost, C 2 12.57 2 $0.80 4 50.27 4 $1.60 6 113.10 6 $2.40 8 201.06 8 $3.20 10 316.16 10 $4.00 Area, A Graph: Pizza Area 300 200 Radius (in.) , r 2 4 6 Area (in2), A 12.57 50.27 113.10 8 201.06 10 316.16 100 2 4 6 8 10 Radius, r The graph does not produce a line. Nonlinear Function. Cost, C Graph: Sauce Cost 6 4 Weight (oz), w 2 Cost, C 4 $1.60 6 $2.40 8 $3.20 10 $4.00 2 2 4 6 8 10 Sauce, w The graph does produce a line. Linear Function. $0.80 YOU TRY IT: Classify the following function as linear or nonlinear. Cutting Paper # of cuts, n 1 Fraction of Original Area 1/2 2 1/4 3 1/8 4 1/16 5 1/32 YOU TRY IT (SOLUTION): # of cuts, Fraction n of Original Area 1 1/2 2 1/4 1.0 Area, A Cutting Paper 0.8 0.6 3 4 5 0.4 0.2 1/8 1/16 1/32 1 2 3 4 5 Cuts, n The graph does not produce a line. Nonlinear Function. REPRESENTING PATTERNS AND NONLINEAR FUNCTIONS: Data from a table can be scrutinize to see if there is any relation or pattern that can help us find what is missing, or complete a table. Ex: What is the pattern we can use to complete the table? Number of Blocks on the edge (x) Total number of Blocks (y) Ordered Pair (x, y) 1 1 (1 , 1) 2 8 (2 , 3) 3 27 (3, 27) 4 ? ? 5 ? ? To answer the question we must take a look at what is happening in the table or the figures: Number of Total Blocks on number of the edge (x) Blocks (y) +1 +1 +1 1 1 2 8 3 27 4 ? 5 ? +7 +19 +37 Notice: although the x increases by 1, the number of blocks is not constant anymore. Taking the info to consideration, we can concentrate on how we manipulate the x to get f(x): f(x) = 1 1 28 3 27 x ? 3 The equation should be f(x) = x f(x) = 4 43 64 f(x) = 5 53 125 Filling in the table: Number of Total Blocks on number of the edge (x) Blocks (y) +1 +1 +1 1 1 2 8 3 27 4 ? 64 5 ? 125 +7 +21 +37 # of Total Blocks Graph: f(x) = x3 Ordered Pair (x, y) (1 , 1) 60 50 (2 , 3) 40 (3, 27) 30 (4, 64) 20 (5, __) 10 2 4 6 8 10 # of Blocks on Edge The graph does not produce a straight line. It is part of the Cubic Functions. YOU TRY IT: Provide the rule that represents the function: x 0 1 2 3 4 y 0 -1 -4 -9 ? YOU TRY IT:(SOLUTION) Looking at the data on the y values we can see that it is not linear, thus we have the following equations to choose from: Y = x2 Y = x3 Y = |x| x 0 1 2 3 4 y 0 -1 -4 -9 ? Notice also that the values are negative, thus: Y = -x2 Y = -x3 Y = -|x| Plugging in numbers we see that y = - x2 should be our equation. Graph: Ordered Pair (x, y) (0 , 0) (1 , -1) (2, -4) (3, -9) (4, -?) f(x) = -x2 The graph does not produce a straight line. It is part of the Square Functions. VIDEOS: Non-Linear functions https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/linear-nonlinearfunctions-tut/v/linear-and-nonlinear-functionsexample-1 https://www.khanacademy.org/math/algebra/line ar-equations-and-inequalitie/linear-nonlinearfunctions-tut/v/linear-and-nonlinear-functionsexample-3 CLASS WORK: Pages: 249 – 251 Problems: As many as it takes to master the concept.