Lecture Guide. Math 90 - Intermediate Algebra. Stephen Toner. Intermediate Algebra, 3rd edition. Miller, O'Neill, & Hyde. Victor Valley College

Transcription

1 Lecture Guide Math 90 - Intermediate Algebra to accompan Intermediate Algebra, 3rd edition Miller, O'Neill, & Hde Prepared b Stephen Toner Victor Valle College Last updated: 7/8/14

2 2.1 The Rectangular Coordinate Sstem Introduction to Graphing The four quadrants: A line is an infinite collection of (,) points, each of which satisf the equation of that line. *Is (3, 4) on the line = 2? *Graph each line: 1. = + 5 (,) coordinates: 2. =

3 3. = Summar: a. When the given line is of the form =, use an - chart with an choices of (usuall = 0,1,2 is best). b. When the given line is of the form a + b = c, also known as standard from, graph using intercepts. c. When the given line is missing a variable, it is either horizontal or vertical = Graph = = 6 7. Find the - and -intercepts of 4 7 =

4 2.2 Slope Formula: m = rise run = *Find the slope of each line: **Parallel lines have the slope. **The slopes of perpendicular lines are of each other. Without graphing, determine wihether the lines through the given pairs of points are parallel, perpendicular, or neither. *Find the slope of the line which runs through the given pair of points: 1. ( 2,3) and (4,1) 4. L L 1 2 : 3, 5 and 1, 2 : 0,4 and 7,2 2. (4,5) and (11, 3) 3. Find the slope of the graphed line (passes through ( 3, 1) and (2,1):

5 2.3 Slope Intercept and Point-Slope Forms *In each equation, identif the slope and the -intercept. 1. = 3 5 *Are the following pairs of lines parallel, perpendicular, or neither? 8. { = = = = { = = { = = = Graph: = = Graph: = 7 18

6 Three Forms of a Line 1. a + b = c standard form d. passes through ( 5,1) and is parallel to the line 3 4 = = m + b slope-intercept form 3. 1 = m( 1 ) point-slope form *Find the equation of the line (in slopeintercept form) which has has the following characteristics. a. m = 2 3 and has -intercept (0,5). b. m = 3 5 and the line passes through ( 1,4). e. passes through ( 6,2) and is perpendicular to the line = c. passes through ( 2,5) and (3,7). 19

7 2.4 Linear Models Ale is a sales representative and earns a base salar of $1000 per month plus a 4% commission on his sales for the month. a. Write a linear equation that epresses Ale s monthl salar in terms of his sales. Let represent the average number of miles driven per ear for passenger cars in the United States since Let =0 represent the ear where corresponds to 1980, =1 corresponds to 1981, and so on. The average earl mileage for passenger cars can be approimated b the equation = where 0. b. Graph the equation. a. Use the linear equation to approimate the average earl mileage for passenger cars in the United States in the ear c. What is the -intercept and what does it represent in the contet of this problem? b. Use the linear equation to approimate the average mileage for the ear 1985, and compare it with the actual value of 9700 mi. d. What is the slope of the line and what does it represent in the contet of this problem? e. How much will Ale make if his sales for a given month are $30,000? c. What is the slope of the line and what does it mean in the contet of this problem? d. What is the -intercept and what does it mean in the contet of this problem? 20

8 3.1 Relations The domain of an epression is the set of values which substituted into the epression. The range of an epression is the set of values which can the epression. To determine the domain from a graph, look at where the graph etends, left to right. To determine the range from a graph, look at where the graph etends verticall. *Find the domain and range of each graph: Eample: = 2 domain: range: To find the domain, start with a "default" domain of and then take awa -values which... ** ** ** *Find the domain of each: f() = g() = 3 f() =

9 3.2 Functions A function is a rule which assigns a -value in the range to each value in its domain. *Do the following relations represent functions? A graph is that of a function if it passes the *Find the domain and range of each: *Are the following graphs those of functions? Notation: f() is pronounced Antime ou see f(), ou can replace it with. 22

10 When we write f(2) = 3, we mean *Find the requested values from the graph: The graph of f() is given. a. Find f(0). Given, { 1. g( 3) = f() = g() = 5 + 7, find the following: h() = d. For what value(s) of is f() = 3? b. Find f(3). c. Find f( 2). e. For what value(s) of is f() = 3? 2. h(10) = f. Write the domain of f(). g. Write the range of f(). 3. f( 3) = The graph of g() is given. 4. g( + 1) = a. Find g( 1). 5. f(w) = b. Find g(1). 6. g( ) = c. Find g(4). 7. h(4) + f(1) = 8. 5 g(2) d. For what value(s) of is g() = 3? e. For what value(s) of is g() = 0? f. Write the domain of g(). g. Write the range of g(). 23

11 *Find the domain of each function. Write the answers in interval notation. m() = Graphs of Basic Functions Graph each of the following. Then state the domain and range of each. 1. f() = Linear Function f(t) = 5 t n(p) = p + 8 p f() = 2 Quadratic Function k(t) = t 5 m() = 1 2 q(t) = t 3 + t 1 3. f() = 3 Cubic Function h ( )

12 4. f() = Absolute Value Function Graph the parabola = Indicate five points on its graph. 5. f() = Square Root Function Graph the cubic function = Indicate three points on its graph. 6. f() = 1 Reciprocal Function 25

13 3.4 The Algebra of Functions 1. Given { f() = 2 + 3, find the following: g() = 2 a. (f + g)() f() = Given { g() = 2 5, find the following: h() = 2 3 a. f g() b. h f() b. (f g)() c. g h(5) c. (f g)( 2) d. g g(3) d. f g() e. g f() f. f g(3) 26

14 The domains of f + g, f g and fg will all be the same (the intersection of their separate domains). The domain of f will be further g restricted so that g() 0. To find the domains of composite functions, compose them and then analze the function that results. 4. Given f() = +3 1 and g() = 4 +2, find... a. the domain of f(). b. the domain of g(). 3. Given f() = 2 and g() = 4, find... a. the domain of f(). c. the domain of f + g, f g and fg. b. the domain of g(). d. the domain of f g (). c. the domain of f + g, f g and fg. d. the domain of f g (). To find the domain of a composite function, find the composition and then analze it. 5. Given f() = 3 and g() = + 7, find the domain of f g(). 27

15 Use the graphs to find the function values: 1. f( 4) 2. f(1) 3. g( 2) 4. g(3) 5. (f + g)(2) 6. (g f)(3) 7. (f g)( 1) 8. (g f)( 4) 9. ( g f ) (0) 10. (f g ) ( 2) 11. (f g ) (0) 12. (g f ) ( 2) 13. g f( 1) 14. f g(0) 15. f g( 4) 16. g f( 4) 17. g g(2) 18. f f( 2) 19. a( 3) 20. a(1) 21. b( 1) 22. b(3) 23. (a b)( 1) 24. (a + b)(0) 25. (b a)(1) 26. (a b)(2) 27. b a (0) 28. a b( 2) 29. a b( 4) 30. b a( 3) 31. ( b ) (3) 32. a (a ) (4) 33. a a ( 2) b 79. The cost in dollars of producing to cars is C() = The revenue received is R() = To calculate profit, subtract the cost from the revenue. a. Write and simplif a function P that represents profit in terms of. b. Find the profit of producing 50 to cars. 28