Summer 2017 Math 108 Section 2.3 67 Section 2.3 Task List Work through each of the following tasks, carefully filling in the following pages in your notebook. Section 2.3 Function Notation and Applications Work through Objective 1 then do problems #1-4 Work through Objective 2 then do problems #5-9 Work through Objective 3 then do problems #10-12
68 Math 108 Section 2.3 Summer 2017
Summer 2017 Math 108 Section 2.3 69 Section 2.3 Function Notation and Applications Section 2.3 Objective 1: Express Equations of Functions Using Function Notation Read through Objective 1 and take notes here: The symbol f() x is read as and is an example of function notation. The symbol f() x represents the value of the variable (output) for a given value of the variable (input). When using function notation, any symbol (usually a letter) can be used to name the function, and any other symbol can be used to represent the independent variable. For example, consider the function y 2x 3, which tells us that the value of the dependent variable (y) is obtained by multiplying the value of the independent variable (x) by 2 and then adding 3. There are infinitely many ways to write this function using function notation. Write down at least 4 different ways of writing y 2x 3using function notation. Be sure to state the function name and the symbol that represents the independent variable.
70 Math 108 Section 2.3 Summer 2017 Read the next page in your e-text and take notes here: Any equation of a function can be written in function notation. Write down the 3-step method used for expressing equations of functions using function notation. Expressing Equations of Functions Using Function Notation Step 1. Step 2. Step 3. Work through the interactive video that accompanies Example 1 and take notes here: Write each function using function notation. Let x be the independent variable and y be the dependent variable. 2 a. y 2x 4 b. y x 0 a. 3x 2y 6 NOW WORK SECTION 2.3 HW EXERCISES #1-4
Summer 2017 Math 108 Section 2.3 71 Section 2.3 Objective 2: Evaluate Functions Read through Objective 2 and take notes here: Work through the interactive video that accompanies Example 2 and take notes here: If f ( x) 4x 5, 2 g( t) 3t 2t 1, and h( r) r 9, evaluate each of the following. a. f (3) b. g( 1) c. h(16) d. f 1 2 NOW WORK SECTION 2.3 HW EXERCISES #5-9
72 Math 108 Section 2.3 Summer 2017 Section 2.3 Objective 3: Graph Simple Functions by Plotting Points Read through Objective 3 and take notes here: Write down the Theorem Theorem One way to sketch the graph of a function given in function notation is by plotting points. Write down the 3-step strategy for graphing simple functions by plotting points. Strategy for Graphing Simple Functions by Plotting Points Step 1. Step 2. Step 3. What is the definition of a linear function? (Linear functions will be discussed in Section 2.4.)
Summer 2017 Math 108 Section 2.3 73 Carefully work through the animation that accompanies Example 3 and graph each function by plotting points. a. f ( x) 2x 1 b. g( x) x 2x 3 2 c. h( x) 2 x 1 NOW WORK SECTION 2.3 HW EXERCISES #10-12
74 Math 108 Section 2.3 Summer 2017
Summer 2017 Math 108 - Section 2.4 75 Section 2.4 Task List Work through each of the following tasks, carefully filling in the following pages in your notebook. Section 2.4 Graphs of Linear Functions Work through Objective 1 then do problems #1-4 Work through Objective 2 then do problems #5-7 Work through Objective 3 then do problems #8-11
76 Math 108 - Section 2.4 Summer 2017
Summer 2017 Math 108 - Section 2.4 77 Section 2.4 Graphs of Linear Functions Section 2.4 Objective 1: Graph Linear Functions by Plotting Points In Section 2.1, you learned how to graph equations by plotting points. See if you can sketch the graph of 2x y 1by plotting points. The equation 2x y 1is an example of a linear equation in two variables. Note that the graph is a line. Write down the definition of a linear equation of two variables below: Linear Equation in Two Variables (Standard Form) Look at the graph of 2x y 1 from the previous page. Does the graph of 2x y 1 represent the graph of a function? Why or why not?
78 Math 108 - Section 2.4 Summer 2017 Write the equation as a function What is the definition of a linear function? Work through Example 1: Graph 4x 3y 3by plotting points. NOW WORK SECTION 2.4 HW EXERCISES #1-2
Summer 2017 Math 108 - Section 2.4 79 Work through the video that accompanies Example 2. Graph 3 f ( x) x 2 by plotting points. 5 NOW WORK SECTION 2.4 HW EXERCISES #3-4 Section 2.4 Objective 2: Graph Linear Functions by Using Intercepts What is the definition of an x-intercept of an equation? What is the definition of an y-intercept of an equation? Given an equation, how to you find the x-intercept? Given an equation, how to you find the y-intercept?
80 Math 108 - Section 2.4 Summer 2017 Work through the video that accompanies Example 3: Graph 2x 5y 8 by using intercepts. Work through the video that accompanies Example 4: Graph 2 f ( x) x 2 by using intercepts. 3 NOW WORK SECTION 2.4 HW EXERCISES #5-7
Summer 2017 Math 108 - Section 2.4 81 Section 2.4 Objective 3: Graph Vertical and Horizontal Lines Recall a linear equation in two variables in standard form is Ax By C where A, B, and C are real numbers and A and B are not both 0. C If B 0, then the standard form equation can be simplified to Ax C or x A intercept. C If A 0, then the standard form equation can be simplified to By C or y b B intercept. a where a is the x- where b is the y- Equations of the form x aare lines. Equations of the form y bare lines. Work through Example 6: Graph y 2. Also, work through Example 7: Graph x 4 Sketch both lines on the same grid. NOW WORK SECTION 2.4 HW EXERCISES #8-11
82 Math 108 - Section 2.4 Summer 2017
Summer 2017 Math 108 - Section 2.5 83 Section 2.5 Task List Work through each of the following tasks, carefully filling in the following pages in your notebook. Section 2.5 Linear Equations in Two Variables Work through Objective 1 then do problems #1-7 Work through Objective 2 then do problems #8-11 Work through Objective 3 then do problems #12-15 Work through Objective 4 then do problems #16-24 Work through Objective 5 then do problems #25-28
84 Math 108 - Section 2.5 Summer 2017
Summer 2017 Math 108 - Section 2.5 85 Section 2.5 Linear Equations in Two Variables Section 2.5 Objective 1: Find the Slope of a Line If the graph of a line is rising or increasing, then the slope of the line is. If the graph of a line is falling or decreasing, then the slope of the line is. If the graph of a line is staying the same or constant then the slope of the line is. Draw three different lines so that one line has positive slope, one line has negative slope, and one line has zero slope. You may want to watch the video for a brief overview of the concept of slope and take notes here:
86 Math 108 - Section 2.5 Summer 2017 Complete the definition of slope below: Definition Slope Given two points,, and, x y x y, on the graph of a line, the slope m of the line containing the 1 1 2 2 two points is given by the formula m = where x1 x 2. Work through Example 1: Sketch the line containing the points 2,1 and 3,5. Then find the slope.
Summer 2017 Math 108 - Section 2.5 87 Work through the video that accompanies Example 2: Sketch the line containing the points 4,5 and 3, 4. Then find the slope. Does it matter if we switch the order of the two points from Example 2 to find the slope? Try finding the slope of the line from Example 2 by switching the order of the two points. That is, find the slope of the line containing the points 3, 4 and 4,5. Do you get the same answer? Work through Example 3: Sketch the line containing the points 3,1 and 5,1. Then find the slope. NOW WORK SECTION 2.5 HW EXERCISES #1-5
88 Math 108 - Section 2.5 Summer 2017 There are many ways to express the equation of a line. Write down the Slope-Intercept Form of a line in the box below. Slope-Intercept Form You may want to watch the video to see how the slope-intercept form is derived from the slope formula. Write your notes here.
Summer 2017 Math 108 - Section 2.5 89 Work through Example 4: Find the slope and y-intercept of the line 6x 5y 10. Work through the video that accompanies Example 5: Find the slope and y-intercept of the line 4x 2y 14. NOW WORK SECTION 2.5 HW EXERCISES #6-7
90 Math 108 - Section 2.5 Summer 2017 Section 2.5 Objective 2: Graph a Line Using the Slope and a Point Carefully work through the animation on graphing a line using the slope and a point and sketch the two lines found in this animation. a) y 2x 3 b) 3x 2y 4 Work through Example 6: Graph the line with slope 1 3 m that passes through the point 1,2.
Summer 2017 Math 108 - Section 2.5 91 Work through the video that accompanies Example 7: Graph the line with slope through the point 1, 4. Also, find three more points on the line. m 2 that passes 3 NOW WORK SECTION 2.5 HW EXERCISES #8-9
92 Math 108 - Section 2.5 Summer 2017 Work through the interactive video that accompanies Example 8: Graph slope and the y-intercept. 3 f ( x) x 2 using the 5 Which of the following is correct? 3 a. Slope ; y-intercept 2 5 b. Slope 2; y-intercept c. Slope d. Slope 3 ; y-intercept 2 5 3 5 3 ; y-intercept 2 5 Which of the following is a correct way to rewrite the slope? (There are two correct answers!) 3 a. Slope ; y-intercept 2 5 3 b. Slope ; y-intercept 2 5 3 c. Slope ; y-intercept 2 5 Now sketch the function 3 f ( x) x 2 5. NOW WORK SECTION 2.5 HW EXERCISES #10-11
Summer 2017 Math 108 - Section 2.5 93 Section 2.5 Objective 3: Determine the Relationship between Two Lines Read through your e-text to help fill in the blanks below: Slopes and y-intercepts can be used to determine the relationship between two lines. Parallel lines have the same but different. Coinciding lines have the same and the same. Perpendicular lines intersect each other at. If two lines are perpendicular, the product of their slopes is. This will occur if the slopes are opposite reciprocals. Define opposite reciprocal and give two examples of pairs of numbers that are opposite reciprocals. What type of lines only intersects?
94 Math 108 - Section 2.5 Summer 2017 Work through the interactive video that accompanies Example 9: For each pair of lines, determine if the lines are parallel, perpendicular, coinciding, or only intersecting. a. 3y 4x 10 16x 12 y 27 b. 3 y x 7 2 6x 4y 28 c. 2 x y 3 3x 6y 5 d. 5 x 2 y 8 x 3y 7 NOW WORK SECTION 2.5 HW EXERCISES #12-15
Summer 2017 Math 108 - Section 2.5 95 Section 2.5 Objective 4: Write the Equation of a Line from Given Information The next form of a line that we will learn is known as the point-slope form of a line. Watch the video to see how the point-slope form of a line is derived. Write the point-slope form of a line in the box below. Point-Slope Form Work through Example 11: Write the equation of the line with slope point 5,8. Write the line in slope-intercept form. m 2 and passing through the 5 NOW WORK SECTION 2.5 HW EXERCISES #16-19
96 Math 108 - Section 2.5 Summer 2017 Work through the interactive video that accompanies Example 12: Write the equation of the line passing through the points 3, 8 and 5, 2. NOW WORK SECTION 2.5 HW EXERCISES #20-22 Work through Example 13: Write the equation of a vertical line passing through the point 4,10 and then write the equation of the horizontal line passing through the same point. NOW WORK SECTION 2.5 HW EXERCISES #23-24
Summer 2017 Math 108 - Section 2.5 97 Section 2.5 Objective 5: Write Equations of Parallel and Perpendicular Lines Before working through Example 14 and Example 15, define parallel and perpendicular lines: Parallel Lines: Perpendicular Lines: Work through the interactive video that accompanies Example 14: Write the equation of the line that passes through the point 3,1 and is perpendicular to 7x 3y 2. Work through the interactive video that accompanies Example 15: Write the equation of the line that passes through the point 3, 2 and is parallel to y 3x 5. NOW WORK SECTION 2.5 HW EXERCISES #25-28
98 Math 108 - Section 2.5 Summer 2017 Table 2 summarizes the important equations and forms of lines discussed in this section. You should memorize everything in this table. Table 2
Summer 2016 Math 108 - Test 1 99 Test 1 Objectives Section 1.1 Linear Equations Determine if a Given Value is a Solution to an Equation..#1, 2 Solving Linear Equations with Integer Coefficients. #3, 4, 5 Solving Linear Equations Involving Fractions..#6, 7, 8, 9 Solving Linear Equations Involving Decimals. #10, 11 Use Linear Equations to Solve Application Problems. #12, 13, 14, 15 Section 1.2 Linear Inequalities in One Variable Determine if a Given Value is a Solution to an Inequality... #16, 17, 18 Graph the Solution Set of an Inequality on a Number Line..#19, 20, 21 Use Interval Notation to Express the Solution Set of an Inequality.. #22, 23, 24 Solve Linear Inequalities in One Variable... #25, 26, 27, 28, 29 Section 1.3 Compound Inequalities; Absolute Value Equations and Inequalities Solve Absolute Value Equations... #30, 31, 32, 33 Section 1.4 Formulas and Problem Solving Solve a Formula for a Given Variable.. #34, 35, 36, 37, 38 Use Formulas to Solve Application Problems Geometric Applications (Perimeter, Surface Area, Volume).. #39, 40 Money...#41, 42 Uniform Motion D rt..#43, 44, 45 Mixture..#46, 47, 48 Section 2.1 The Rectangular Coordinate System and Graphing Plot Ordered Pairs in the Rectangular Coordinate System... #49, 50, 51 Determine if an Ordered Pair is a Solution to an Equation... #52, 53, 54 Find Unknown Coordinates... #55, 56, 57 Graph Equations by Plotting Points... #58, 59, 60, 61, 62 Find x- and y-intercepts..... #63, 64, 65 Section 2.2 Relations and Functions Identify Independent and Dependent Variables... #66, 67 Find the Domain and Range of a Relation... #68, 69, 70, 71 Determine if Relations are Functions... #72, 73, 74, 75 Determine if Graphs are Functions.. #76, 77, 78, 79 Section 2.3 Function Notation and Applications Express Equations of Functions Using Function Notation... #80, 81, 82 Evaluate Functions.... #83, 84, 85 Graph Simple Functions by Plotting Points.. #86, 87, 88, 89
100 Math 108 - Test 1 Summer 2017 Section 2.4 Graphs of Linear Functions Graph Linear Functions by Plotting Points...... #90, 91, 92 Graph Linear Functions by Using Intercepts.... #93, 94, 95 Graph Vertical and Horizontal Lines.... #96, 97, 98 Section 2.5 Linear Equations in Two Variables Find the Slope of a Line......... #99-103 Determine the Slope and y-intercept given an Equation of a Line.... #104, 105 Graph a Line Using the Slope and a Point..... #106, 107, 108, 109 Determine the Relationship between Two Lines... #110-115 Write the Equation of a Line from Given Information Given the slope and y-intercept........ #116, 117, 118, 119 Given a point and the slope........... #120, 121, 122 Given two points.......#123, 124 Write Equations of Parallel and Perpendicular Lines.... #125, 126, 127, 128