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2.1 Represent Relations and Functions Standards: Anchors: 2.8.8I, 2.8.11RST M11.D.1.1.1, M11.D.1.1.2, M11.D.1.1.3, M11.D.2.1.2, M11.D.3.1.2 OBJECTIVE: TLW write relations as sets, mappings, tables, graphs and equations TLW identify and evaluate functions VOCABULARY Relation: Domain: Range: Function: REPRESENTING RELATIONS A relation can be represented in the following ways: Ordered Pairs (2, 2) (2, 2) (0, 1) (3, 1) Table x y 2 2 2 2 0 1 3 1 Graph Mapping Diagram Input 2 0 3 Output 2 2 1 Function: is a relation for which each input has exactly one output. If any input of a relation has more than one output, the relation is NOT a function. Identify functions 1) Tell whether each relation is a function. Explain. a. Input Output b. Input Output 2 1 3 6 1 0 4 2 1 2 2 0 2 3 Is the relation given by the ordered pairs (5, 2), (3, 1), (0, 0), (0, 2) and (0, 5) a function? Explain. VERTICAL LINE TEST Function Not a function Is the relation represented by the graph a function? Explain. a. b. Classify and evaluate functions Linear function in x-y notation y = mx + b Linear function in function notation f(x) = mx + b Linear function has the variable or variables:x and y to the first power. Tell whether the function is linear. Then evaluate the function when x = 3. a. f(x) = 6x + 10 b. g(x) = 2x2 + 4x 1 Use the vertical line test to tell whether the relation is a function. Tell whether the function is linear. Then evaluate the function when x = 1. 1. f(x) = 2x3 + 6 x 2. g(x) = 4x + 9 Homework: Skill 2.1 2.2 Slope and Graphing Linear Equations Standards: Anchors: 2.8.8DFGHI, 2.8.11EKLNQS, 2.11.8B M11.C.3.1.2, M11.D.2.1.2, M11.D.3.1.1, M11.D.3.2.1, M11.D.3.2.3 OBJECTIVE: TLW calculate slope TLW identify lines as intersecting, parallel, or perpendicular TLW graph lines using the slope-intercept method TLW graph lines using the x-y intercept method TLW graph horizontal and vertical lines. VOCABULARY Slope: of a nonvertical line is the ratio of vertical change (the rise) to horizontal change (the run) Parallel: two lines in a plane never intersect Perpendicular: two lines in a plane intersect to form a right angle Rate of change: or slope is how much one quantity changes, on average relative to the change in another quantity SLOPE OF A LINE Words The slope m of a nonvertical line is the ratio of __________ change (the rise) to _______ change (the run). Algebra m Graph y2 y1 x2 x1 Example 1 Find slope What is the slope of the line passing through the points (1, 3) and (6, 7)? Let (x1, y1) (1, 3) and (x2, y2) (6, 7). CLASSIFICATION OF LINES BY SLOPE Positive slope Rises from left to right Negative Zero Undefined slope slope slope Falls Horizontal Vertical from left to right Classify lines using slope Without graphing, tell whether the line through the given points rises, falls, is horizontal, or is vertical. a. (6, 2), (1, 3) b. (2, 1), (2, 2) Complete the following exercises. 1. Find the slope of the line passing through the points (4, 2) and (7, 9). 2. Without graphing tell whether the line through the points (3, 2) and (1, 4) rises, falls, is horizontal, or is vertical. SLOPES OF PARALLEL AND PERPENDICULAR LINES Consider two different nonvertical lines l1 and l2 with slopes m1 and m2. Parallel lines The lines are parallel if and only if they have the______ Perpendicular lines The lines are perpendicular if and only if their slopes are________________ Classify parallel and perpendicular lines Tell whether the lines are parallel or perpendicular. Line 1: through (3, 1) and (2, 5) Line 2: through (3, 4) and (3, 1) Tell whether the lines are parallel, perpendicular, or neither. Line 1: through (1, 0) and (3, 4) Line 2: through (24, 6) and (22, 5) VOCABULARY y-intercept Slope-intercept form Standard form of a linear equation x-intercept USING SLOPE-INTERCEPT FORM TO GRAPH AN EQUATION Step 1 Write the equation in ______________ form by solving for y. Step 2 _______ the y-intercept b and use it to plot the point (0, b) where the line crosses the y -axis. Step 3 Identify the ________ m and use it to plot a second point on the line. Step 4 ________ a line through the two points. Graph an equation in slope-intercept form Graph y = 32 x + 1. USING STANDARD FORM TO GRAPH AN EQUATION Step 1 Write the equation in standard form. Step 2 Identify the x-intercept by letting ___ = 0 and solving for ____. Use the x-intercept to plot the point where the line crosses the______. Step 3 Identify the y-intercept by letting ____ = 0 and solving for ____. Use the y-intercept to plot the point where the line crosses the _______. Step 4 Draw a line through the two points. Graph an equation in standard form Graph 2x + 3y = 12. HORIZONTAL AND VERTICAL LINES Horizontal lines The graph of y = c is the horizontal line through (____,___). Vertical lines The graph of x = c is the vertical line through (____,____). Graph horizontal and vertical lines a. Graph y = 1 b. Graph x = 2. Graph the equation. 1. y = 2x + 2 3. 4x + 2y = 8 4 2. y = 3 x 4 4. 5x + 3y = 15 5. y = 4 Homework: Skill 2.2 6. x = 2 2.3 Write Equations of Lines Standard: Anchor: 2.8.11L M11.D.3.2.2 OBJECTIVE: TLW write linear equations when given: slope and a point two points parallel to a given line and a point perpendicular to a given line and a point Given slope m and y-intercept Use slope-intercept form: y = mx + b Given slope m and a point(x,y) Use point-slope form: y – y1 = m (x – x1) Given points (x1,y1) and (x2, y2) Find slope then use point-slope form WRITING AN EQUATION OF A LINE 1. find the slope of the line 2. use the slope and the point to set up equation using the point slope form 3. change formula to standard form Write an equation given the slope and y-intercept Write an equation of the line shown. Write an equation given the slope and a point Write an equation of the line that passes through (2, 1) and has a slope of 2. Write equations of parallel or perpendicular lines Write an equation of the line that passes through (1, 1) and is (a) parallel to, and (b) perpendicular to, the line y = 2x + 3. Write an equation given two points Write an equation of the line through (3, 1) and (2, 3). Write an equation of the line. 1. 2. 3. Through (1, 5) with a slope of 2 Through (2, 3) and (a) parallel and (b) perpendicular to y = 4x 6 4. Through (6, 2) and (3, 2) Homework: Skill 2.3 2.4 Draw Scatter Plots and Best Fitting Lines Standards: 2.6.8AC, 2.6.11ACD, 2.8.11A Anchors: M11.D.1.1.1, M11.E.1.1.1, M11.E.4.2.2 OBJECTIVE: TLW identify correlations TLW find a line of best fit with and without a calculator VOCABULARY Scatter plot- is a graph of a set of data pairs (x,y) Positive correlation – the y value tends to increase as x increases Negative correlation – the y value tends to decrease as x increases Correlation coefficient – denoted r is a number from -1 to 1 that measures how well a line fits a set of data points. r is near 1 points lie near line with positive slope r is near -1 points lie near line with negative slope r is near 0 points do not lie near any line. Best-fitting line – is the line that lies as close as possible to all the data points if correlation coefficient for a set of data is near ±1 the data can be reasonably modeled by a line. Estimate correlation coefficients For each scatter plot, describe the correlation shown and tell whether the correlation coefficient is closest to 1, 0.5, 0, 0.5, or 1. a. b. Solution a. The scatter plot shows a ____________ correlation. So, the best estimate given is r = ____. b. The scatter plot shows a ___________ correlation. So, r is between ___ and ___ but not too close to either one. The best estimate given is r = ____. APPROXIMATING A BEST-FITTING LINE Step 1 Draw a _________ of the data. Step 2 Sketch the _____ that appears to follow most closely the trend given by the data points. There should be about as many points ______ the line as _______ it. Step 3 Choose _________ on the line, and estimate the coordinates of each point. Step 4 Write an _________ of the line that passes through the two points from Step 3. Approximating a best-fitting line The table below gives the number of people y who attended each of the first seven football games x of the season. Approximate the best-fitting line for the data. x 1 2 3 4 y 722 763 772 826 1. Draw a ___________. 5 6 7 815 857 897 Be sure that about the same number of points lie above your line of fit as below it 2. Sketch the best-fit line. 3. Choose two points on the line. For the scatter plot shown, you might choose(1, _____ ) and (2, _____ ). 4. Write an equation of the line. Use a line of fit to make predictions Use the equation of the line of best fit from Example 2 to predict the number of people that will attend the tenth football game. For each scatter plot (a) tell whether the data has positive correlation, negative correlation, or no correlation, and (b) tell whether the correlation coefficient is closest to 1, 0.5, 0, 0.5, or 1. 1. 3. The table gives the average class score y on each chapter test for the first six chapters x of the textbook. x 1 2 3 4 5 6 y 84 83 86 88 87 90 a. Approximate the best-fitting line for the data. b. Use your equation from part (a) to predict the test score for the 9th test that the class will take. Homework: Skill 2.4 2.5 Use Absolute Value Functions Standards: 2.5.11ABCD, 2.8.11RST Anchors: M11.A.2.2.1 Objectives: TLW graph absolute value functions TLW graph using transformations, translations, and reflections VOCABULARY Absolute value function Vertex of an absolute value graph Transformation Translation Reflection The highest or lowest point on the graph of an absolute value function is called the _____________________________. The vertex of the graph y | x | is (___, ___). TRANSFORMATIONS OF GENERAL GRAPHS Graph functions of the form y = a | x | 1 Graph (a) y = 3 x and (b) y = 2 | x |. Compare each graph with the graph of y | x |. Graph the function. Compare the graph with y = | x |. 1. y = 3 | x | Graph a function of the form y a | x h | k Graph y = 3 | x 2 | 1. Compare the graph with the graph of y = | x |. Write an absolute value equation Write an equation of the graph shown. 2. Graph the function y graph with the graph of y = | x |. 1 2 | x 1 | 2. Compare the Write an equation of the graph shown. Homework: Skill 2.5 2.6 Other Functions Standards: 2.5.11ABCD, 2.8.11RST Anchors: M11.D.1.1.1, M11.D.1.1.2, M11.D.1.1.3 OBJECTIVE: TLW evaluate, graph and write piecewise (compound) functions TLW evaluate, graph and write step functions Vocabulary: Compound function Step function Compound Functions Use the function below to evaluate and graph: 3x 2 f ( x) x 2 Find f(-3) x0 x0 f(2) f(0) Graph the function Write a function for the following problem: A silk screen shop has the following price schedule for silk-screen t-shirts. An initial charge of $25 to create the silk-screen $10.50 per shirt for orders of 25 or fewer shirts $9.75 per shirt for orders of more than 25 shirts Step Functions Use the function below to evaluate and graph: 10 4 f ( x) 1 3 x 1 1 x 2 2 x4 x4 Find f(-3) f(2) f(10) Graph the function Homework: Skill 2.6 2.7 Graph Linear Inequalities in Two Variables Standard: Anchor: 2.8.8EFG, 2.8.11KN M11.D.2.1.2 Objective: TLW graph linear inequalities in two variables with and without a graphing calculator Check whether the ordered pairs (a) (3, 2) and (b) (1, 4) are solutions of 4x + 2y > 6. Check whether the ordered pair is a solution of 2x y 8. 1. 2. (6, 2) (3, 1) GRAPHING A LINEAR INEQUALITY To graph a linear inequality in two variables, follow these steps: Step 1 Graph the boundary line for the inequality. Step 2 Test a point ________ the boundary line to determine whether it is a solution of the inequality. Graph a linear inequality with one variable Graph y < 1 in a coordinate plane. Graph a linear inequality with two variables Graph 3x 2y < 6 in a coordinate plane. Graph an absolute value inequality Graph y > 3 x 1+ 2 in a coordinate plane. Graph the inequality in a coordinate plane. 3. x < 2 5. 9x + 3y > 9 4.y x + 2 6. y 2 |x + 2| l GRAPHING INEQUALITIES ON TI-84 Use Apps – Inequalities Homework: Skill 2.7