Section 1 Solving Linear Equations and Inequalities A. Evaluating Expressions Examples Evaluate the following if a = 7, b =, and c = ( a + c ) + b. a + b c B. The Distributive Property Try the Following Simplify each expression. (x y) 8( x + y). (x y) 6( x y) C. Solving Equations Try the Following Solve for x. (x + ) ( x 1) = 78. 1 x =. 8x 9 + = D. Solving Formulas Try the Following Solve for the variable specified 1 Mm d = at ; a. F = G ; m r E. Inequalities Try the following Solve and graph the inequalities y x + + 6 8 x. 1 >. 1 x 7. y + 9 or 18 > y 10 Precalculus Chapter 1 Page 1
D. Absolute Value Equations Try the following Solve for x. 8x = 1. x + 7 = E. Absolute Value Inequalities Try the following solve and graph. x 7. x 1 F. x + 0 HW Worksheet Precalculus Chapter 1 Page
Section Linear Relations and Functions I. Relations and Functions A. Definitions Relation. Domain. Range. Function B. Examples Determine whether each relation is a function. a) Mappings: 7 0 1-9 b) Tables: x -7 - y -1-9 - 0 c) Ordered Pairs: {(,),(,),(0,7),(0,9) }. Graphs: 1) ) ) Note: Vertical Line Test Precalculus Chapter 1 Page
C. Function Values and Notation Function Notation a) y = x 8 vs. f ( x) = x 8 b) Function notation has the advantage of clearly identifying the dependent variable f ( x ), formerly known as y, while at the same time telling you x is the independent variable and that the function itself is called f. c) Function notation allows you to be less wordy. Instead of asking What is the value of y that corresponds to x =? you can ask What is f ( )? :. Examples a) If f ( x) = x, find f (). b) If h( x) = 0.x x.7, find h (6). c) If f x =, find ( ) ( ) x f t. d) Extra: If g a ( ) a 6 =, find g( b + ). II. Linear Equations A. Forms... B. Slope and intercepts Slope. x-intercept. y-intercept Precalculus Chapter 1 Page
C. Examples Find the intercepts and slope of the following: a) x y = 16 b) 1 x y = c) y = x. Find the equation of a line with the following description: a) Slope of and goes through the point of (-1,6). b) Goes through the points (,-1) and (,-8). c) Goes through the points (,-1) and (8,-1). II. Parallel and Perpendicular A. Parallel B. Perpendicular C. Examples: Find an equation of a line that is parallel to x y = 7 and goes through the point (-,).. Find an equation of a line that is perpendicular to y = x + and goes through the point (,1). Precalculus Chapter 1 Page
III. Modeling Linear Functions A. The table below shows the approximate percent of students who sent applications to two colleges in various years since 198. Draw a scatter plot, best fit line, and find an equation that would predict the approximate percent of students who sent applications to two colleges in various years since 198. What percent of students sent applications in 199? What type of correlation does this data have? Years since 198 0 6 9 1 1 Percent 0 18 1 1 1 1 IV. Special Functions A. Direct Variation B. Constant Function C. Identity Function D. Absolute Value Function E. Piecewise Function F. Step Function (Greatest Integer Function) G. Examples Graph y = x +. Graph x 1, if x f ( x) = 1, if x > Homework: Worksheets. Graph y = x Precalculus Chapter 1 Page 6
Section Systems of Linear Equations and Inequalities Solving systems Graphically Types and and Examples x + y = x y = 0 y = x +. 9x + y = 1 Algebraically Elimination x + y = x y = 0 1 - - - - -1 1 6-1 - - - -. 7 x y = 17 x + y = 1 1 - - - - -1 1 6-1 - - - - Substitution y = x + 9x + y = 1. x + y = x y = 0 Technology x + y = x y = 0. y = x + 9x + y = 1 Precalculus Chapter 1 Page 7
Solving Systems Equations in Three Variables There are 9,000 seats in a sports stadium. Tickets for the seats in the upper level sell for $, the ones in the middle level cost $0, and the ones in the bottom level are $ each. The number of seats in the middle and bottom levels together equals the number of seats in the upper level. When all of the seats are sold for an event, the total revenue is $1,19,00. How many seats are there in each level? Inequalities x + y x y 0 6-6 8 10 - -8-10 -1 x + y 6. x y x + y 1 10 8 6. y < x + 9x + y > 1 1 - - - - -1-1 1 6 - - - - -10-8 - - 6 8 10 - - -8-10 Precalculus Chapter 1 Page 8
Linear Programming Graph the following constraints. Then find the maximum and minimum values of the function f ( x, y) = x y. x y x + y 8 6 6 8 10 1. A landscaping company has crews who rake leaves and mulch. The company schedules hours for mulching jobs and hours for raking jobs. Each crew is scheduled for no more than raking jobs per day. Each crew s schedule is set up for a maximum of 8 hours per day. On the average, the charge for raking a lawn is $0 and the charge for mulching is $0. Find a combination of raking leaves and mulching that will maximize the income the company receives per day from one of its crews. 6 Profit Function: Constraints: a. b. c. d. 6 8 Homework Worksheet Precalculus Chapter 1 Page 9