QUIZ AND TEST INFORMATION: The material in this chapter is on Quiz 3 and Exam 2. You should complete at least one attempt of Quiz 3 before taking Exam 2. This material is also on the final exam and used on future exams. TEXT INFORMATION: The material in this chapter corresponds to the following sections of your text book: 1.7, 1.8, 1.9, 2.1, 2.2, 2.3, 2.4, 2.6. Please read these sections and complete the assigned homework from the text that is given on the last page of the course syllabus. LAB INFORMATION: Material from these sections is used in the following labs: Graphing Techniques and the Box Lab. This information is also needed for future labs. Inequalities Lines and Circles Introduction to Functions Assignments 1.7-1.9, 2.1-2.4, 2.6 Labs Graphing Techniques The Box Problem Quiz 3 Test 2 43
Section 1: Solving Inequalities Definition: When we are asked to solve an inequality, we are asked to find the values for the variable that make the inequality true. This set of values is often referred to as the s s of the inequality. We often use i n to describe the solution set of an inequality. Hint for solving inequalities: When you multiply or divide both sides of an inequality by a negative number, then you. Example 1: Solve 2-3x 5 and graph the solution set. 44
Example 2: Solve 4 2x + 2 10 and graph the solution set. Fact: A product of two numbers is positive if both pieces are positive or both pieces are negative. A product of two numbers is negative if one number is negative and the other is positive. Example: 45
Example 3: Solve (x - 5)(x + 2) > 0. Fact: A fraction is positive if the numerator and the denominator are positive, or if the numerator and the denominator are negative. A fraction is negative if the numerator is positive and the denominator is negative, or vice versa. A fraction is zero if and only if. Example: 46
x + 1 Example 4: Solve 0. x 1 Fact: If x and y are nonzero real numbers such that 1 1 <, then x > y. x y Example: 47
1 1 Example 5: Solve <. x 2 3x 9 Example 6: Solve 0 1/x 1/3. 48
Example 7: Find the real numbers a and b such that, if 2 < x < 4, then a < 1/(x - 6) < b. Other examples and notes: 49
Section 2: Solving Inequalities with Absolute Value Example 1: If x < 3, then x is units from 0 on the number line. This means that x is an element of the set. On the number line this looks like: Example 2: If x > 3, then x is units from 0 on the number line. This means that x is an element of the set. On the number line this looks like: Note: Absolute Value Inequalities 1. u < a is equivalent to < u <. 2. u > a is equivalent to u < or u >. 3. Similarly with less than or equal to. Example 3: Translate x - 1 < 5 into a sentence about distance on the number line. 50
Example 4: Solve x - 1 < 5 and graph the solution set. Example 5: Translate x 1 5 into a sentence about distance on the number line. Example 6: Solve x - 1 5 and graph the solution set. 51
Example 7: Solve for x. x - 2 + 2 < 3 Example 8: Solve for x. x - 3 2 Example 9: Find a and b. If x - 1 < 3, then a < x + 4 < b. 52
Example 10: Express the fact that x differs from 2 by less then ½ as an inequality involving an absolute value. Solve for x. Other example and notes: 53
Section 3: Basics of Lines Note: A measure for the steepness of a line is called. We usually use the letter to stand for the slope of a line. Definition: Two special cases of lines are v and h lines. V l have slope that is and H l have slope that is. Note: A vertical line through the point (a, b) will have equation. Note: A horizontal line through the point (a, b) will have equation. Example 1: Give the equation of the vertical line passing through the point (-5, 2). Example 2: Give the equation of the line with undefined slope passing through the point (4, -3). 54
Example 3: Give the equation of the horizontal line passing through the point (-5, 2). Example 4: Give the equation of the line with slope zero passing through the point (4, -3). Slope Formula: Given two points (x 1, y 1 ) and (x 2, y 2 ) on a nonvertical line, the slope, m, of the line is given by the formula: Example 5: Draw some examples of lines with positive slope and some lines with negative slope. 55
Example 6: Find the slope of the line passing through the points (1, -2) and (-2, 5). Definition: The y-intercept of a line is. We usually use the letter to stand for the y-coordinate of the y-intercept of a line (meaning that the intercept is the point (0, )). Example 7: Find the y-intercept of the line 5x + 6y = 2. Example 8: Find the y-intercept of the line f(x) = 5x + 17. 56
Forms for equations of lines: In the following, m stands for the slope of the line, b stands for the y-coordinate of the y-intercept of the line, and (x 1,y 1 ) is an arbitrary point on the line. 1. Standard Form: 2. Point-Slope Form: 3. Slope-Intercept Form: Note: Linear functions are functions of the form. Linear functions are polynomials of degree. The domain of any linear function is. Example 9: Find the equation of the line passing through the points (7, 2) and (5, 1). 57
Example 10: Find the equation of the line with slope -3 passing through the point (2, 0). Other examples and notes: Other examples and notes: Other examples and notes: 58
Section 4: Parallel and Perpendicular Lines Example 1: Draw some examples of parallel and perpendicular lines. Note: Two distinct lines are parallel if and only if they have the same. Note: Two lines are perpendicular if and only if the product of their slopes is (neither line of which is vertical or horizontal). Example 2: Are the following lines parallel? 2x + 3y = 10, 8x + 12y = 13 59
Example 3: Find the equation of the line that is parallel to y = -2x + 1 and passes through the point (0, -1). Example 4: If a given line has equation y = 1/2 x + 5, then a line perpendicular to this line has the equation y = x + 10. Example 5: Find the equation for the line that is perpendicular to y = -2x + 1 and passes through the point (3, 4). 60
Example 6: Find an equation for a line that contains the point (1, -2) and is perpendicular to 4x +3y - 6 = 0. Other examples and notes: 61
Section 5: Circles Definition: A circle is a set of points that are a fixed distance r (the ) from a fixed point (h, k) (the ) in the Cartesian plane. Forms for equations of circles: 1. General Form: 2. Standard Form: Completing the square: 1. Step 1: Take the coefficient of x 2. Step 2: Divide this number by 2 3. Step 3: Square this number Example 1: What number should be added to x 2 + 6x to complete the square? 62
Example 2: Complete the square on x 2-6x = 5. Example 3: Find the center and radius of the circle given by the following equations. 1. x 2 +y 2 =1 Center=(, ) Radius= 2. (x-1) 2 +(y+3) 2 =16 Center=(, ) Radius= 3. x 2 +y 2-2x+6y-6=0 Center=(, ) Radius= 63
Other examples and notes: 64
Section 6: Symmetry Definition: A graph is symmetric with respect to the y-axis if. Note: An algebraic check to see if the graph of a function is symmetric with respect to the y-axis is. Example 1: Draw a graph that is symmetric with respect to the y-axis. Definition: A function is if its graph is symmetric with respect to the y-axis. Definition: A graph is symmetric with respect to the origin if. 65
Note: An algebraic check to see if the graph of a function is symmetric with respect to the origin is. Example 2: Draw a graph that is symmetric with respect to the origin. Definition: A function is if its graph is symmetric with respect to the origin. Definition: A graph is symmetric with respect to the x-axis if. Example 3: Draw a graph that is symmetric with respect to the x-axis. 66
Example 4: Is f(x) = x 2 odd, even, or neither? Example 5: Is f(x) = x 3 odd, even, or neither? Example 6: Is f(x) = x 2-4 odd, even, or neither? 67
Example 7: Determine the symmetry algebraically. F(x) = 2x 2 + 2 Example 8: Given the points (1, 5) and (1, -5), what sort of symmetry do the points have? Example 9: Make a complete graph (intercepts, symmetry, and basic shape) of f(x) = x 2, g(x) = x 3, and h(x) = x 2-4. Work for f(x) = x 2 : 68
Graph of f(x) = x 2 Work for g(x) = x 3 : Graph of g(x)=x 3 69
Work for h(x) = x 2 4: Graph of h(x) = x 2-4 Other examples and notes : 70
Section 7: Definition of Function Definition: Let X and Y be two non-empty sets. A function from X to Y is a. Example 1: Do the following represent functions? 1. {(2,3), (4,5), (6,7)}, because. 2. {(2,3), (7,9), (2,4)}, because. 3. {(Bob, $100), (Mary, $200), (Sue, $300)}, because. 4. {(Bob, $100), (Mary, $100), (Sue, $300)}, because. 5., because. 6. 71
, because. 7. y = x 2, because. Sketch the graph here. 8. y = x 3, because. Sketch the graph here. 72
9. x = y 2, because. Sketch the graph here. 10. Other examples: 73
Definition: The vertical line test says. Example 2: Does the following graph represent a function? Other examples and notes: 74
Section 8: Domain and Range Definition: Given a function, f(x), from a non-empty set X to a nonempty set Y, the set X is called the of the function. Definition: Given a function, f(x), the range of f(x) is. Example 1: Find the domain and range of the following functions. 1. {(3,2), (4,2), (2,3)} Domain= Range= 2. {(Bob, $100), (Mary, $200), (Sue, $300)} Domain= Range= Example 2: Find the domain of the following functions. 1. F(x) = x 2 + 1 Domain= 2. G(x) = any polynomial Domain= 75
3. f 2x ( x) = 2 9 x 4. f ( x) = 5 2x 76
Example 3: Find the domain and range of the following functions by considering their graphs. 1. F(x) = x 2 Domain= Range= 2. G(x)=x 3 Domain= Range= 77
Other examples and notes : 78
Section 9: Operations with functions (f + g)(x)= (f - g)(x)= (f g)(x)= (f/g)(x) =, g(x) 0 Example 1: Find (f + g)(x) when f(x) = x + 1 and g(x) = x - 2. Example 2: If G(x) = 2x 2 + x + 1, evaluate the following. 1. G(1)= 2. G(-x)= 3. G(x)= 4. G(x+1)= 79
5. G(x)+G(1)= Definition: The Average Rate of Change of a Function (also known as the Difference Quotient) is. Notation: Some commonly used notation for the average rate of change of a function y = f(x): Formula: If c is in the domain of a function f, then the average rate of change from c to x is defined by: 80
Formula: Another way to find the average rate of change of a function f is: Example 3: Evaluate the average rate of change when F(x) = x 2 + 1 and c = 2. 81
Example 4: Find the difference quotient of f(x) = -x 2 + 3x - 2. Other examples and notes: 82
Section 10: More on Graphs of Functions Definition: A function f(x) is increasing on an open interval if for any x 1 and x 2 in the open interval with, then. Draw an example: Definition: A function f(x) is decreasing on an open interval if for any x 1 and x 2 in the open interval with, then. Draw an example: 83
Definition: A function f(x) is constant on an open interval if for any x 1 and x 2 in the open interval. Draw an example: Example 1: On what open intervals is f(x) = x 2 increasing? Decreasing? Constant? 84
Maxima and Minima: 2 Example 2: Will f(x) = -x have a max, min, or neither? Other examples and notes: 85
Section 11: Graphs of Special Functions Example 1: Draw graphs of the following functions (some examples are families of functions). Give the domains and ranges. List any other special properties that the graphs might have. 1. Constant Functions 86
2. Linear Functions 87
3. The Identity Func tion 88
4. The Square Function 89
5. The Square Root Func tion 90
6. The Cube Function 91
7. The Cube Root Func tion 92
8. The Reciprocal Function 93
Other examples and notes: 94
Section 12: Piecewise Defined Functions Note: We need to remember the graphs of functions with their domains and ranges to graph piecewise defined functions. Example 1: f ( x ) = x x 3 if x 0 if x < 0 Example 2: Given the previous function, what is f(3)? What is f(-5)? 95
Example 3: The Greatest Integer Function Other examples and notes: Other examples and notes: 96