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:

1. Linear : Graph is nonvertical line or part of a nonveritcal line 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 = -mx + b y = n 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 = 𝒙 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. Pizza Area Sauce Cost Radius (in.) , r Area (in2), A 2 12.57 4 50.27 6 113.10 8 201.06 10 316.16 Weight (oz), w Cost, C 2 $0.80 4 $1.60 6 $2.40 8 $3.20 10 $4.00

The graph does not produce a line. Nonlinear Function. Pizza Area Radius (in.) , r Area (in2), A 2 12.57 4 50.27 6 113.10 8 201.06 10 316.16 Area, A 300 200 100 2 4 6 8 10 Radius, r The graph does not produce a line. Nonlinear Function.

The graph does produce a line. Linear Function. Sauce Cost Weight (oz), w Cost, C 2 $0.80 4 $1.60 6 $2.40 8 $3.20 10 $4.00 Cost, C 6 4 2 2 4 6 8 10 Sauce, w The graph does produce a line. Linear Function.

Fraction of Original Area YOU TRY IT: Classify the following function as linear or nonlinear. Cutting Paper # of cuts, n Fraction of Original Area 1 1/2 2 1/4 3 1/8 4 1/16 5 1/32

The graph does not produce a line. Nonlinear Function. YOU TRY IT (SOLUTION): Cutting Paper # of cuts, n Fraction of Original Area 1 1/2 2 1/4 3 1/8 4 1/16 5 1/32 1.0 Area, A 0.8 0.6 0.4 0.2 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.

Number of Blocks on the edge (x) Total number of Blocks (y) 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) 2 8 (2 , 3) 3 27 (3, 27) 4 ? 5

Number of Blocks on the edge (x) Total number of Blocks (y) To answer the question we must take a look at what is happening in the table or the figures: Number of Blocks on the edge (x) Total number of Blocks (y) 1 2 8 3 27 4 ? 5 +1 +7 +1 +19 +1 +37 Notice: although the x increases by 1, the number of blocks is not constant anymore.

The equation should be f(x) = x3 Taking the info to consideration, we can concentrate on how we manipulate the x to get f(x): f(x) = 1  1 2  8 3  27 x ? The equation should be f(x) = x3 f(x) = 4  43 64 f(x) = 5  53 125

Number of Blocks on the edge (x) Total number of Blocks (y) Filling in the table: Number of Blocks on the edge (x) Total number of Blocks (y) 1 2 8 3 27 4 ? 5 +1 +7 +1 +21 +1 +37 64 125

# of Total Blocks # of Blocks on Edge Graph: f(x) = x3 Ordered Pair (x, y) (1 , 1) (2 , 3) (3, 27) (4, 64) (5, __) 60 50 # of Total Blocks 40 30 20 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.

Provide the rule that represents the function: x y 1 - 1 2 - 4 3 - 9 4 YOU TRY IT: Provide the rule that represents the function: x y 1 - 1 2 - 4 3 - 9 4 ?

YOU TRY IT:(SOLUTION) x y 1 - 1 2 - 4 3 - 9 4 ? 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| 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/linear-equations-and-inequalitie/linear-nonlinear-functions-tut/v/linear-and-nonlinear-functions-example-1 https://www.khanacademy.org/math/algebra/linear-equations-and-inequalitie/linear-nonlinear-functions-tut/v/linear-and-nonlinear-functions-example-3

Problems: As many as it takes to master the concept. CLASS WORK: Pages: 249 – 251 Problems: As many as it takes to master the concept.