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Intermediate Algebra Chapter 2 •The •Coordinate Plane •and •Functions Babe Didrikson Zaharias, athlete • “The formula for success is simple: Practice and concentration, then more practice and more concentration.” Section 2.1 • • • • Rectangular Coordinate system Plotting points Intercepts Graphing Calculator Graphing Calculator Keys • • • • • • • 2nd QUIT , Window, Arrow Keys Y= Clear ZOOM 6 – Standard ZOOM 8 – Integer TRACE ZOOM IN ZOOM OUT Calculator Keys • [VARS] • [Y-VARS] • Evaluating function • Try Y = 3x – 2 for x =5 Intermediate Algebra 2.1 • The Graph of an Expression Repeated Evaluation of expression • Enter expression in [Y=] • [VARS][Y-VARS] • [1:Function] • [1:Y][ENTER] Section 2.2 Evaluate Expression • Enter expression in Y screen • And produce table nd • 2 TBLSET nd • 2 TABLE Graphing calculator keys • • • • • • [Y=] [Window] [Graph] [Trace] [Zoom] [Zoom Integer] Setting Window • By Hand • Zoom • 6:Zstandard • 8:Zinteger • X[-9.4,9.4] Y[-10,10] friendly window • Zbox • Zoom In • Zoom Out • Z Decimal Modeling • Algebraic Models of situations are not perfect. • Values of dates and variables need to be examined carefully • Models can give predictions • Some models are better than others Section 2.2 • Relations and Functions • *********************** Jackie Joyner-Kersee - athlete •“It is better to look ahead and prepare than to look back and regret.” Relation • A set of Ordered Pairs. • {1,2,(3,4)} • {(2,3),(2,4)} Domain • The set of first components of ordered pairs. • {(1,2),(3,4)} • Domain = {1,3} Range • The set of second components of ordered pairs. • {(1,2),(3,4)} • Range = {2,4} Function • Is a relation in which no two ordered pairs have the same first components. • {(1,2),(3,4)} Vertical Line Test • The graph of a relation represents a function if and only if no vertical line intersects the graph at more than one point Interval Notation • • • • (2,5) (2,5] [2,5] [2,5) [2, ) (,2] (2, ) (,2) Section 2.2 • Function Notation • and • Evaluation Functional Notation • f(x) read “f of x” • Name of the function is f • x is the domain element • f(x) is the value of the range Calculator evaluation • • • • • • Table Y= YVARS Program Evaluate Plug In Store feature Intermediate Algebra 2.3 •Analysis •of •Functions Lou Holtz – football coach •“No one has ever drowned in sweat.” Analysis of Functions Odell • • • • • • Maximums Minimums Intercepts - zero Points of Intersection-zero Domain Range Absolute Maximum • Y coordinate of the highest point of the graph of the function Absolute Minimum • Y-coordinate of the lowest point of the graph of the function. Local Maximum • Highest point in a “neighborhood” • Local Minimum • Lowest point in a “neighborhood.” X intercept • A point at which the graph intersects the x-axis. • At this point y = 0 • [CALC]2:zero Y-Intercept • A point at which the graph intersects the y-axis. • At this point x = 0. • Calculator – many times can be found using trace Points of Intersection • The point(s) at which the two graphs of two function on the same set of axes intersect each other. Calculator Keys • • • • • • 2nd CALC ZERO MINIMUM MAXIMUM INTERSECT VALUE • Study Groups are Useful Unknown author • “Today, be aware of how you are spending your 1,440 beautiful moments, and spend them wisely.” Intermediate Algebra Chapter 2 • Properties • of • Lines Intermediate 2.3 •Linear Equations •In •Two Variables Def: Linear Equation • A linear equation in two variables is an equation that can be written in standard form ax + by = c where a,b,c are real numbers and a and b are not both zero. Def: Solution of linear equation in two variables • A solution of a linear equation in two variables is a pair of numbers (x,y) that satisfies the equation. • Ex:{(3,4)} Def: Intercepts • y-intercept – a point where a graph intersects the y-axis. • x-intercept is a point where a graph intersects the x-axis. Procedure to find intercepts • To find x-intercept • 1. Replace y with 0 in the given equation. • 2. Solve for x • To find y-intercept • 1. Replace x with 0 in the given equation. • 2. Solve for y Find solutions to Equations with 2 variables • 1. Choose a value for one of the variables • 2. Replace the corresponding variable with you chosen value. • 3. Solve the equation for the other variable. Horizontal Line •y = constant • Example: y = 4 • y-intercept (0,4) • Function – no x intercept Vertical Line • x = constant • Example x = -5 • x-intercept (-5,0) • No y intercept • Not a function Intermediate 2.4 •Slope of a Line Objective: • Given two points, determine the slope of a line. Slope Slope rise y2 y1 y m run x2 x1 x Horizontal line • y = constant • Slope is 0 • Examples: y = 5 • y = -3 • Can be done with calculator. Vertical Line • x=constant • Undefined slope • Examples: • x =2 • x = -3 • Not graphed by calculator Slope Intercept Form for equation of Line • y=mx+b Slope is m y-intercept is (0,b) Using Slope Intercept form to graph a line • 1. Write the equation in form y=mx+b • 2. Plot y intercept (0,b) • 3. Write slope with numerator as positive or negative • 3. Use slope – move up or down from y intercept and then right- plot point. • 4. Draw line through two points. Problem • The percentage B of automobiles with airbags can be modeled by the linear function B(t)-5.6t –3.6, where t is the number of years since 1990. • What is the slope of the graph of B? • Answer is 5.6 Fred Couples – Professional Golfer • “When you’re prepared you’re more confident: when you have a strategy you’re more comfortable.” Intermediate – 2.5 •Applications •Of •Slope Objectives: • Determine if two lines are parallel. • Determine if two lines are perpendicular. Def: Parallel Lines • Two distinct non-vertical lines are parallel if and only if they have the same slope. • Two distinct vertical lines are parallel. Def 1: Perpendicular Lines • Two distinct lines are perpendicular if and only if the product of their slopes is –1. • A vertical line and horizontal line are perpendicular. Def 2: Perpendicular Lines • The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. • If slope is a/b, slope of perpendicular line is –b/a. Helen Keller – advocate for he blind • “Alone we can do so little, together we can do so much.” Intermediate Algebra 2.5 •Equations •of •Lines Objective • Use slope-intercept form to write the equation of a line. y=mx+b • Write the equation of a line given the slope and the y intercept. • Line slope is 2 and y intercept (0,-3) • y=2x-3 y=mx+b • Write the equation of a line given the slope and one point. • Slope of 2 and point (1,3) • y=2x+1 Point-slope form of Linear equation given slope of m & pt ( x1 , y1 ) y y1 m( x x1 ) Objective: Write equation of a line given the slope and one point • Problem: slope of –3 through (2,-4) • Answer: y=-3x+2 Objective – Write equation of line given two points • Given points (-3,6),and (9,-2) • Find slope • Slope is –2/3 • Answer: y=(-2/3)x+4 Objective: Write equation of a line in slope-intercept form that passes through (4,-1) and is parallel to y=(-1/2)x+3 • y=(-1/2)x+1 Intercepts Form for equation of a line. • a is the x-intercept • b is the y-intercept x y 1 a b Pop Warner – football coach •“You play the way you practice.”