Unit 1: Function Families GPS Algebra Unit 1: Function Families

Unit 1: Function Families Function Notation Graphing Basic Functions Function Characteristics

Function Notation Use and Purpose of Function Notation Presentation of Functions in Tables, Mappings, Graphs, and Algebraic Function Notation Determining Whether a Relation is a Function Introduction of Domain and Range

Methods of Representing Relations Tables Mappings x y -1 1 2 4 -1 1 2 1 4 Graphs Function Notation f(x) = x2 Read: “f of x equals x squared” Function notation is an efficient way to talk about functions

Is a Relation a Function? A relation is a function if there is only one output for any of its inputs (i.e. an x input can only lead to one y output) Table/Mapping Check – repeats in domain (x)? x y -1 1 2 4 x y 2 8 3 6 5 10 1 NOT A FUNCTION FUNCTION Graph Check – vertical line hits one or fewer points on the graph? NOT A FUNCTION FUNCTION

Domain and Range of a Function Inputs in a Table or Mapping Independent Variable (x) on a Graph How far left and right my function goes x Range Outputs in a Table or Mapping Dependent Variable (y) on a Graph How far up and down my function goes f(x)

Domain and Range of a Function x f(x) -1 1 2 4 -1 1 2 Domain: {-1, 0, 1, 2} Range: {0, 1, 4} 1 4 With continuous functions, domain and range are expressed in interval notation. Domain: All Real #s The farthest left/right the graph goes Range: The farthest up/down the graph goes

Using Function Notation Function notation is used to represent relations which are functions. Finding f(-2), for example, is the same as evaluating an expression for x = -2. Example: f(x) = 3x – 1; find f(-2) Solution: f(-2) = 3(-2) – 1 = -6 – 1 = -7 The same substitution process can be used to complete a function table. x f(x) = 3x + 5 f(x) -2 -1 1 3(-2) + 5 = -6 + 5 -1 3(-1) + 5 = -3 + 5 2 The inputs x are the domain; the outputs f(x) are the range 3(0) + 5 = 0 + 5 5 3(1) + 5 = 3 + 5 8

Graphing Basic Functions Graphing parent functions for linear, absolute value, quadratic, square root, cubic, and rational functions Introduction to transformations

Linear & Absolute Value Functions f(x) = x f(x) = |x| Domain: All Real #s Range: All Real #s Domain: All Real #s Range: y is greater than or equal to 0

Quadratic & Square Root Functions f(x) = x2 f(x) = Domain: All Real #s Range: y is greater than or equal to 0 Domain: x is greater than or equal to 0 Range: y is greater than or equal to 0

Cubic & Rational Functions f(x) = x3 f(x) = Domain: All Real #s Range: All Real #s Domain: All Real #s except x cannot equal 0 Range: All Real #s except y cannot equal 0

Parent Graphs on the Move Translation up Translation down f(x) = x3 + 1 f(x) = |x| – 3 Domain: All Real #s Range: All Real #s Domain: All Real #s Range: y is greater than or equal to -3

Vertical Stretch and Shrink Coefficients determine the shape of a graph; a coefficient outside the function results in a vertical stretch or shrink 3f(x) is vertically stretched by 3 ½ f(x) is vertically shrunk by 2 f(x) = 3x2 Vertical stretch by 3 (rises three times as fast) f(x) = ½x2 Vertical shrink by 2 (rises half as fast) Domain: All Real #s; Range: y is greater than or equal to 0 Domain: All Real #s; Range: y is greater than or equal to 0

Reflections of a Function The sign of a coefficient indicates whether it is reflected across the x-axis or y-axis -f(x) is reflected across the x-axis f(-x) is reflected across the y-axis f(x) = -|x| f(x) = Domain: x is greater than or equal to 0 Range: y is less than or equal to 0 Domain: All Real #s Range: y is less than or equal to 0

Multiple Transformations These transformations can occur together with changes to the coefficient and what is added or subtracted to the function Parent function: x2 (quadratic) f(x) = - ½ (x )2 + 3 Reflect down Shift up 3 Vertical shrink by 2 Domain: All Real #s Range: y is less than or equal to 3

Multiple Transformation Practice Write the function for the graph below Graph the following function f(x) = 3(x )2 – 3 f(x) = -2(x )2 + 4 Domain: All Real #s Range: y is greater than or equal to 3 Domain: All Real #s Range: y is less than or equal to 4

Function Characteristics Analyzing graphs by determining domain, range, zeros, intercepts, intervals of increase and decrease, maximums and minimums, and end behavior

Does this function EVER stop?! Analyzing Functions Domain: how far left and right? Range: how far up and down? Zeros: x-intercepts – where does the function intersect the x-axis? Intercepts: zeros and y-intercepts Intervals of decrease and increase: where does the function go up and where does it go down? Maximums and Minimums: what’s the highest and/or lowest the function goes? End Behavior: as inputs approach infinity, what happens to the function? Does this function EVER stop?!

Analyzing Functions: Domain and Range LINEAR ABSOLUTE VALUE Domain: all real numbers in the input Range: all real numbers in the output Domain: all real numbers in the input Range: lowest point is -1; goes up forever  y is greater than or equal to -1

Analyzing Functions: Domain and Range SQUARE ROOT QUADRATIC Domain: furthest left is 0; goes right forever  x is greater than or equal to 0 Range: lowest point is -3; goes up forever  y is greater than or equal to -3 Domain: all real numbers in the input Range: lowest point is +1; goes up forever  y is greater than or equal to 1

Analyzing Functions: Domain and Range CUBIC RATIONAL Domain: all real numbers in the input Range: all real numbers in the output Domain: left and right forever, but skips over 2  All real's except 0 Range: up and down forever, but skips over -1  All real’s except -1

Analyzing Functions: Zeros and Intercepts LINEAR ABSOLUTE VALUE Zeros: one x-intercept: (-2, 0) y-intercept: (0, -1) Zeros: two x-intercepts: (-1 , 0) and (1, 0) y-intercept: (0, -1)

Analyzing Functions: Zeros and Intercepts QUADRATIC SQUARE ROOT Zeros: no real zeros (the graph never intersects the x-axis) y-intercept: (0, 1) Zeros: one x-intercept: (2, 0) y-intercept: (0, -3)

Analyzing Functions: Zeros and Intercepts CUBIC RATIONAL Zeros: (1, 0) y-intercept: (0, -1) Zeros: (1, 0) y-intercept: None

Analyzing Functions: Intervals of Decrease and Increase LINEAR ABSOLUTE VALUE Intervals of decrease: x is less than and greater than 0 Intervals of increase: none Intervals of decrease: x is less than 0 Intervals of increase: x is greater than 0

Analyzing Functions: Intervals of Decrease and Increase QUADRATIC SQUARE ROOT Intervals of decrease: the left half of the function, x is less than 0 Intervals of increase: the right half of the function, x is greater than 0 Intervals of decrease: none (the entire function is uphill) Intervals of increase: the entire domain of the function x is greater than 0

Analyzing Functions: Intervals of Decrease and Increase CUBIC RATIONAL Intervals of decrease: none (the entire function is uphill) Intervals of increase: the entire domain of the function x is less than and greater than 0 Intervals of decrease: the entire domain of the function, x is less than and greater than 0 Intervals of increase: none (the entire function is downhill)

Analyzing Functions: Maximums and Minimums Maximum: the highest point of the graph For example, a cannon is shot into the air. The maximum is where it changes from going up to going down (i.e. the highest it goes) Minimum: the lowest point of the graph For example, a stock broker is watching the market searching for a good time to buy Alpha-Bit stocks. She looks for a stock that appears to have reached a low cost and is about to begin to increase in value.

Analyzing Functions: Maximums and Minimums LINEAR ABSOLUTE VALUE Maximums: none Minimums: none Maximums: none Minimums: (2, -1)

Analyzing Functions: Maximums and Minimums QUADRATIC SQUARE ROOT Maximums: none Minimums: (0, 1) Maximums: none Minimums: (0, -3)

Analyzing Functions: Maximums and Minimums CUBIC RATIONAL Maximums: none Minimums: none Maximums: none Minimums: none

Analyzing Functions: End Behavior As x approaches - (forever left): does the function approach - (forever down) or  (forever up)? As x approaches  (forever right): does the function approach - (forever down) or  (forever up)? Sample notation: as x  -, f(x)  

Analyzing Functions: End Behavior LINEAR ABSOLUTE VALUE End Behavior: as x  -, f(x)   left arm up as x  , f(x)  - right arm down End Behavior: as x  -, f(x)   left arm up as x  , f(x)   right arm up

Analyzing Functions: End Behavior QUADRATIC SQUARE ROOT End Behavior: as x  -, f(x)   left arm up as x  , f(x)   right arm up End Behavior: as x  , f(x)   right arm up

Analyzing Functions: End Behavior CUBIC f(x) End Behavior: as x  -, f(x)  - left arm down as x  , f(x)   right arm up