CHAPTER 1 RELATIONS AND LINEAR FUNCTIONS

Cartesian Coordinate Plane

VOCABULARY RELATION – SET OF ORDERED PAIRS RELATION – SET OF ORDERED PAIRS – example: {(4,5), (–2,1), (5,6), (0,2)} DOMAIN – SET OF ALL X’S DOMAIN – SET OF ALL X’S – D: {4, –2, 5, 0} RANGE – SET OF ALL Y’S RANGE – SET OF ALL Y’S – R: {5, 1, 6, 2} A relation can be shown by a mapping, a graph, equations, or a list (table). A relation can be shown by a mapping, a graph, equations, or a list (table).

Function A function is a special type of relation. – By definition, a function exists if and only if every element of the domain is paired with exactly one element from the range. – That is, for every x-coordinate there is exactly one y-coordinate. All functions are relations, but not all relations are functions.

One-to-one Mapping – example: {(4,5), (–2,1), (5,6), (0,2)}

Function Example – B: {(4,5), (–2,1), (4,6), (0,2)} – Not a function because the domain 4 is paired with two different ranges 5 & 6

Vertical Line Test The vertical line test can be applied to the graph of a relation. If every vertical line drawn on the graph of a relation passes through no more than one point of the graph, then the relation is a function.

Graphing by domain and range Y=2x+1 – Make a table to find ordered pairs that satisfy the equation – Find the domain and range – Graph the ordered pairs – Determine if the relation is a function

More Vocab. Function Notation A function is commonly denoted by f. In function notation, the symbol f (x), is read "f of x " or "a function of x." Note that f (x) does not mean "f times x." The expression y = f (x) indicates that for every value that replaces x, the function assigns only one replacement value for y. f (x) = 3x + 5, Let x = 4 also written f(4) – This indicates that the ordered pair (4, 17) is a solution of the function.

Function Examples If f(x) = x³ - 3, evaluate: If f(x) = x³ - 3, evaluate: – f(-2) – f(3a) If g(x) = 5x 2 - 3x+7, evaluate: If g(x) = 5x 2 - 3x+7, evaluate: – g(4-2a) – g(-3c)

EVALUATING A LINEAR FUNCTION The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit, f, that are equivalent to a given number of degrees Celsius, C On the Celsius scale, normal body temperature is 37°C. What is the normal body temp in degrees Fahrenheit? On the Celsius scale, normal body temperature is 37°C. What is the normal body temp in degrees Fahrenheit?

Find the domain of each function F(x) = x 3 +5x F(x) = x 3 +5x x 2 -4x x 2 -4x The denominator can’t be zero. G(x) = 1 G(x) = 1 √x-4 √x-4 Radicandcan’t be negative.

CHAPTER 1.2 Composition of Functions

Sum: (f+g)(x)=f(x) + g(x) Difference: (f-g)(x)=f(x) - g(x) Product: (f*g)(x)=f(x) * g(x) Quotient: (f/g)(x)=f(x) / g(x) Operations with Functions

[f o g](x)=f[g(x)] f[g(x)] means to substitute the function g(x) wherever you see an x in f(x) Composition of Functions

CHAPTER 1.3 Graphing Linear Equations

USE INTERCEPTS TO GRAPH A LINE X – INTERCEPT SET Y=0 SET Y=0 Y – INTERCEPT SET X=0 SET X=0 PLOT POINTS AND DRAW LINE EX: - 2X + Y – 4 = 0

SLOPE CHANGE IN Y OVER CHANGE IN X CHANGE IN Y OVER CHANGE IN X RISE OVER RUN RISE OVER RUN RATIO RATIO STEEPNESS STEEPNESS RATE OF CHANGE RATE OF CHANGE FORMULA FORMULA

Slope Positive Slope Negative Slope 0 Slope Undefined Slope

USE SLOPE TO GRAPH A LINE 1. PLOT A GIVEN POINT 1. PLOT A GIVEN POINT 2. USE SLOPE TO FIND ANOTHER POINT 2. USE SLOPE TO FIND ANOTHER POINT 3. DRAW LINE 3. DRAW LINE EX: DRAW A LINE THRU (-1, 2) WITH SLOPE -2

SLOPE-INTERCEPT FORM y = mx + b y = mx + b m IS SLOPE m IS SLOPE b IS Y-INTERCEPT b IS Y-INTERCEPT

CHAPTER 1.4/1.5 Writing Linear Equations & Writing Equations of Parallel and Perpendicular Lines

POINT-SLOPE FORM FIND SLOPE FIND SLOPE PLUG IN PLUG IN ARRANGE IN SLOPE INTERCEPT FORM ARRANGE IN SLOPE INTERCEPT FORM

EX: WRITE AN EQUATION OF A LINE THRU (5, -2) WITH SLOPE -3/5 EX: WRITE AN EQUATION OF A LINE THRU (5, -2) WITH SLOPE -3/5 EX: WRITE AN EQUATION FOR A LINE THRU (2, -3) AND (-3, 7) EX: WRITE AN EQUATION FOR A LINE THRU (2, -3) AND (-3, 7)

INTERPRETING GRAPHS WRITE AN EQUATION IN SLOPE- INTERCEPT FORM FOR THE GRAPH WRITE AN EQUATION IN SLOPE- INTERCEPT FORM FOR THE GRAPH

REAL WORLD EXAMPLE As a part time salesperson, Dwight K. Schrute is paid a daily salary plus commission. When his sales are $100, he makes $58. When his sales are $300, he makes $78. Write a linear equation to model this. Write a linear equation to model this. What are Dwight’s daily salary and commission rate? What are Dwight’s daily salary and commission rate? How much would he make in a day if his sales were $500? How much would he make in a day if his sales were $500?

LINES PARALLEL SAME SLOPE SAME SLOPE VERTICAL LINES ARE PARALLEL VERTICAL LINES ARE PARALLELPERPENDICULAR OPPOSITE RECIPROCALS (flip it and change sign) OPPOSITE RECIPROCALS (flip it and change sign) VERTICAL AND HORIZONTAL LINES VERTICAL AND HORIZONTAL LINES

Write an equation for the line that passes through (3, -2) and is perpendicular to the line whose equation is y = -5x + 1 Write an equation for the line that passes through (3, -2) and is perpendicular to the line whose equation is y = -5x + 1 Write an equation for the line that passes through (3, -2) and is parallel to the line whose equation is y = -5x + 1 Write an equation for the line that passes through (3, -2) and is parallel to the line whose equation is y = -5x + 1

CHAPTER 1.7 Piecewise Functions

Different equations used for different intervals of the domain Different equations used for different intervals of the domain

Piecewise Functions F(x)= { 1 if x≤-2 F(x)= { 1 if x≤-2 2+x if -2<x ≤3 2x if x>3

Step Function Looks like a set of stairs Looks like a set of stairs Breaks in the graph of the function Breaks in the graph of the function

Greatest Integer Function Type of step function Type of step function Means the greatest integer Means the greatest integer not greater than x. Example: [[8.9]]=8

ABSOLUTE VALUE FUNCTION V-shaped V-shaped PARENT GRAPH (Basic graph) PARENT GRAPH (Basic graph)

Examples F(x)=2│x │-6 F(x)=2│x │-6 F(x) = {2x+1 if x<0 F(x) = {2x+1 if x<0 2x-1 if x≥0 F(x) = [[x-1]] F(x) = [[x-1]] F(x) = { x+3 if x≤0 F(x) = { x+3 if x≤0 3-x if 1<x ≤3 3x if x>3

CHAPTER 1.8 GRAPHING INEQUALITIES

BOUNDARY EX: y ≤ 3x + 1 EX: y ≤ 3x + 1 THE LINE y = 3x + 1 IS THE BOUNDARY OF EACH REGION THE LINE y = 3x + 1 IS THE BOUNDARY OF EACH REGION SOLID LINE INCLUDES BOUNDARY SOLID LINE INCLUDES BOUNDARY____________________________ DASHED LINE DOESN’T INCLUDE BOUNDARY DASHED LINE DOESN’T INCLUDE BOUNDARY

GRAPHING INEQUALITIES 1. GRAPH BOUNDARY (SOLID OR DASHED) 1. GRAPH BOUNDARY (SOLID OR DASHED) 2. CHOOSE POINT NOT ON BOUNDARY AND TEST IT IN ORIGIONAL INEQUALITY 2. CHOOSE POINT NOT ON BOUNDARY AND TEST IT IN ORIGIONAL INEQUALITY 3. TRUE-SHADE REGION WITH POINT 3. TRUE-SHADE REGION WITH POINT FALSE-SHADE REGION W/O POINT

On calculator Enter slope-int form under “y=“ Enter slope-int form under “y=“ Scroll to the left to select above or below Scroll to the left to select above or below Zoom 6 Zoom 6

GRAPH THE FOLLOWING INEQUALITIES x – 2y < 4

Review

Quiz F(x)=│x+3│ F(x) = { x+3 if x≤0 3-x if 1<x ≤3 3x if x>3

CHAPTER 1.6 LINEAR MODELS

Prediction line SCATTER PLOT – GRAPH WITH MANY ORDERS PAIRS SCATTER PLOT – GRAPH WITH MANY ORDERS PAIRS LINE OF BEST FIT – LINE DRAWN THROUGH DATA THAT BEST REPRESENTS IT LINE OF BEST FIT – LINE DRAWN THROUGH DATA THAT BEST REPRESENTS IT

MAKE A SCATTER PLOT APPROXIMATE PERCENTAGE OF STUENTS WHO SENT APPLICATIONS TO TWO COLLEGES IN VARIOUS YEARS SINCE 1985 YEARS SINCE %

LINE OF BEST FIT SELECT TWO POINTS THAT APPEAR TO BEST FIT THE DATA SELECT TWO POINTS THAT APPEAR TO BEST FIT THE DATA IGNORE OUTLIERS IGNORE OUTLIERS DRAW LINE DRAW LINE

PREDICTION LINE FIND SLOPE FIND SLOPE WRITE EQUATION IN SLOPE-INTERCEPT FORM WRITE EQUATION IN SLOPE-INTERCEPT FORM

INTERPRET WHAT DOES THE SLOPE INDICATE? WHAT DOES THE SLOPE INDICATE? WHAT DOES THE Y-INT INDICATE? WHAT DOES THE Y-INT INDICATE? PREDICT % IN THE YEAR 2010 PREDICT % IN THE YEAR 2010 HOW ACCURATE ARE PREDICTIONS? HOW ACCURATE ARE PREDICTIONS?

Regression Regression Line  Line of best fit Linear correlation coefficient (r) – The closer the value of r is to 1 or -1, the closer the data points are to the line.