Algebra II Notes Unit Two: Linear Equations and Functions

Transcription

1 Syllabus Objectives:.1 The student will differentiate between a relation and a function.. The student will identify the domain and range of a relation or function.. The student will derive a function rule from a given set of data using correct function notation..4 The student will sketch the graph of a function. Relation: a set of ordered pairs Domain: the set of input values x in a relation Range: the set of output values y in a relation Note: x is called the independent variable, and y is called the dependent variable Ex: Use the relation: 1,, 0,4, 0,, 1,. State the domain and range. Solution: Domain 1, 0,1 ; Range,, 4 Ex: Popcorn Prices: Small 00 Medium 4 00 Large 5 00 How much would ten large popcorns cost? (1, 5 00 ) Cost equals the number multiplied by $5 (, ) (, ) C = n $5 (10,?) C = 5n C = 5(10) = $50 The people in front of you in line all buy some large popcorn: (, ), (, ) and (1, 5 00 ). You order large popcorns and the popcorn guy says That will be $18. Is everything functioning here? Function: a special type of relation in which each input has exactly one output Ex: Is the relation a function? If so, state the domain & range. a) Input Output b) Input Output No, because the input 4 has two output values. Yes, every input has exactly one output. Page 1 of 0 Domain: 1,, 5, 7 ; Range:,4,6

2 Graphs of Functions: given the graph, we can use the vertical line test to determine if a relation is a function. Vertical Line Test: a graph is a function if all vertical lines intersect the graph at most once Ex: Which of the graphs is a function? Not a function fails the vertical line test Function passes the vertical test Function Notation: f x (Read f of x, or the value of f at x.) Ex: Evaluate f 4 for the function f x x 1. Substitute 4 x into the function. f Graphing a Function: Create a table of values. Choose values for the independent variable and evaluate the function at these values. Plot the points to graph the function. Ex: Graph the function f ( x) x 1 Step One: Choose values for x. (It is best to choose two negative values, zero, and two positive values.) x f(x) Step Two: Evaluate the function at these values and complete the table. x f(x) Step Three: Plot the points on a coordinate plane and graph the function. Page of 0

3 Writing a Function Rule: To write a function rule for a set of data, look for a pattern in the data. Ex: Write a function rule for the given set of data. g x x Solution: 1 x g(x) Functions in Real Life: Ex: The profit for the school play is $4 per ticket minus $80 for the expense to build the set. There are 00 seats in the theater. a) Write an equation that represents the profit (p) for n tickets sold. Solution: p 4n 80 b) Describe the domain of the function. Solution: nn, 70n00 (Read: All n in the set of integers such that n is between and including 70 and 00. ) c) Make an input-output table. (Note: input values may vary) Solution: n p d) Describe the range of the function. Solution: 0 p 90 You Try: It takes 150 hours to resurface 0 miles of road. a) Write a linear function for resurfacing d miles of road in t hours. b) Find the length of the road done in 500 hours. QOD: Explain using the definition of a function why the vertical line tests determine whether a graph is a function. Page of 0

4 Sample CCSD Common Exam Practice Question(s): 1. Which relation is a function?. What is the domain of the following relation? {(5, ),( 1,7),( 4,1)} A. { 4, 1,5} B. { 5,1,4} C. {,7,1} D. { 1, 7,} Page 4 of 0

5 Sample SAT Question(s): Taken from College Board online practice problems. 1. Which of the following equations expresses y in terms of x for each of the four pairs of values shown in the table above? (A) y 5x 7.5 (B) y 5.5x (C) y 5.5x 7.5 (D) y 7.5x (E) y 7.5x 5.5 Grid-In. The price of a certain item was $10 in 1990 and it has gone up by $ per year since If this trend continues, in what year will the price be $100? Page 5 of 0

6 Syllabus Objectives.6 The student will calculate the slope of a line..7 The student will recognize slope as a rate of change of one variable in terms of another. Slope of a Line: the ratio of the vertical change of a line to the horizontal change of a line Note: Another name for slope is rate of change. Formula for Slope: given two points,, and, Slope = m = x y x y, you can use the formula to calculate slope 1 1 vertical change horizontal change = rise change in y = run change in x = y y 1 x x y y = 1 1 x x1 Ex: Find the slope of the line that passes through the points 5,7 and 4, m or m Finding the slope of a line given the graph: pick two points on the line and find the ratio of the vertical change to the horizontal change. Note: UP is a positive vertical change, and DOWN is a negative vertical change. RIGHT is a positive horizontal change, and LEFT is a negative horizontal change. Special Cases (Horizontal and Vertical Lines): Ex: Find the slope of the line passing through the points 4, and 4,. m 5 undefined (vertical line) Ex: Find the slope of the line passing through the points 1, and, 0 m 0 (horizontal line) 1 Page 6 of 0

7 Ex: Use two points on each line to find the slopes of the lines. O e) e Solution: f) OF Solution: g) GO Solution: h) h Solution: d C B a m undefined 0 m 1 4 m m a) a Solution: b) OB Solution: c) CO Solution: d) d Solution: h G 0 m m 4 4 m m e O F Steepness of Lines: the line with the slope of the greatest absolute value is the steeper line Ex: Which line is steeper? A line with a slope of or a line with a slope of 5? Slope as Rate of Change Solution: The line with the slope of Ex: Find the rate of change in dollars per hour if an employee makes $4 in 4 hours and $10 in 0 hours $ hr Page 7 of 0

8 Challenge Problem: Ex: Find the value of y so that the line that passes through the points 4, and 4, y has a slope of 1. m y y y 1 y 4 8 y 4 y 8 You Try: Sketch a line with a a) positive slope c) negative slope b) zero slope d) slope that is undefined QOD: How can you tell from the graph of a line if the slope is positive, negative, zero, or undefined? Page 8 of 0

9 Sample CCSD Common Exam Practice Question(s): 1. Use the graph below. What is the slope of the line? A. B. C. D.. At 10:00 a.m., a thermometer reads 79 F. At 4:00 p.m., it reads 10 F. What is the average rate of change in the temperature? A. 4F per hour B. F 4 per hour C. 4F per hour D. 4F per hour Page 9 of 0

10 Sample SAT Question(s): Taken from College Board online practice problems. 1. A boat costs x dollars, and this cost is to be shared equally by a group of people. In terms of x, how many dollars less will each person contribute if there are 4 people in the group instead of? (A) 1 x (B) 4 x (C) x (D) 7 x 1 (E) 7x. For the first part of his bike trip, Dag rode down a hill at x miles per hour for t hours. For the rest of his trip, Dag rode up a hill at half that speed for twice as long. What was Dag s average speed, in miles per hour, for his entire trip? (A) 1 x (B) x (C) 4 x (D) 5 6 x (E) x tx 1 y. The equation above is the equation of a line in the xy-plane, and t is a constant. If the slope of the line is 10, what is the value of t? Grid-In Page 10 of 0

11 Syllabus Objective.10 The student will sketch the graphs of linear and absolute value equations and inequalities. Slope-Intercept Form of an Equation of a Line: y mx b ; m = slope, and b = y-intercept y-intercept: the value b in the ordered pair 0,b ; where the line intersects the y-axis Graphing a Line in Slope-Intercept Form Ex: Graph the line 1 y x Step One: Plot the y-intercept. b Step Two: From the point plotted in Step One, go down 1 and to the right to plot the next point. Note: You could have also plotted a point going up 1 and to the left Step Three: Connect the points to draw the line. Standard Form of an Equation of a Line: Ax By C, where both A and B are not zero. x-intercept: the x-coordinate of the point where the line intersects the x-axis. Note: The line will intersect the x-axis when the y-coordinate is 0. y-intercept: the y-coordinate of the point where the line intersects the y-axis. Note: The line will intersect the y-axis when the x-coordinate is 0. Sketching the graph of a line in standard form can be done by finding the intercepts. Ex: Graph the line 4xy 1. Step One: Find the x-intercept (let y 0 ). 4x 0 14x 1 x Step Two: Find the y-intercept (let x 0 ). 40y 1y 1 y 4 Step Three: Plot the intercepts and connect to draw the line. Page 11 of 0

12 Special Cases: Horizontal and Vertical Lines Let a and b be real numbers. The line y a is a horizontal line. (Slope is 0, y-intercept is a.) The line x b is a vertical line. (Slope is undefined, x-intercept is b.) Ex: Graph the line x. Ex: Graph the line y 4. Slope-Intercept Form in Real-Life: Used when there is one rate in the problem. Ex: Between 1990 & 1998 Riverton s population decreased by about 150 people per year. The population was 75,450 in Write a linear equation to model the situation, and then use it to estimate the population in (Let t represent the number of years since 1990.) Use y mx b, or P mt b: m is the rate of change and b is the initial value people m 150 b people in 1990 year Solution: P 150t t 7 P There were approximately 74,400 people in Riverton in ; 1997: Standard Form in Real-Life: Used when there are two rates in the problem. Ex: Student tickets to a football game cost $.00, and adult tickets cost $6.00. Ticket sales at Friday s game totaled $000. Write an equation that shows the number of student tickets (s) and adult tickets (a) that could have been sold. Graph the equation and describe the combinations of student and adult tickets that satisfy the model. Solution: s6a 000 s a Any ordered pair as, that lie on the line graphed above are combinations that satisfy the model. Note: 0 s 1000 and 0 a 500. Page 1 of 0

13 You Try: a) Graph the line y x 4. What is the slope? What is the y-intercept? b) Graph the line x5y 10. What are the x- and y-intercepts? QOD: What do the slope and y-intercept represent in a linear model of a real-life application? Sample CCSD Common Exam Practice Question(s): Which graph represents the equation 9x4y 6? Page 1 of 0

14 Sample SAT Question(s): Taken from College Board online practice problems. If x and y are positive integers, what are all the solutions x, y of the equation xy 11? (A) 1, 4 only (B),1 only (C) 1, 4 and, (D) 1, 4 and,1 (E), and,1 Page 14 of 0

15 Syllabus Objectives:.5 The student will write the equation of a line or linear function from a given set of data..8 The student will use slopes to classify lines as parallel, perpendicular, or neither..9 The student will write equations of parallel and perpendicular lines..11 The student will explore relationships among families of lines. Writing the Equation of a Line Given the Graph: Use slope-intercept form y mx b, where m = slope, and b is the y-intercept. Ex: Write the equation of the line shown in the graph. Step One: Find the y-intercept (b). b Step Two: Find the slope by choosing two points on the line and finding the ratio of the vertical change to the horizontal change (rise over run). m 1 Step Three: Write the equation of the line using y mx b. y x Writing the Equation of a Line Given the Slope and a Point on the Line: Use point-slope form, where m = slope, and, y y m x x 1 1 x y is a point on the line. 1 1 Ex: Write the equation of the line in slope-intercept form with a slope of 4 that passes through the point 1, 5. Step One: Substitute the point and slope into point-slope form. y 54x 1 Step Two: Rewrite in slope-intercept form be solving for y. y 54x4 y 4x 9. Writing the Equation of a Line Given Two Points: Use point-slope form y y mx x 1 1 Ex: Write the equation of a line in standard form that passes through the points, and 6,1. Step One: Find the slope. y1 y m x x Step Two: Substitute the slope and either point into point-slope form. y x Step Three: Rewrite in standard form. 1 y 6 x y x x y 4 Page 15 of 0 1

16 Parallel Lines: lines in the same plane that do not intersect Note: Parallel lines have equal slopes. Perpendicular Lines: lines in the same plane that intersect at a right angle Note: Perpendicular lines have slopes that are opposite reciprocals. Writing Equations of Parallel/Perpendicular Lines: Ex: Write the equation of the line that is parallel to the line point,5 in standard form. y x 1 that passes through the Step One: Determine the slope of the given line. m Note: Because parallel lines have equal slopes, this is also the slope of the new line. Step Two: Use the slope and the given point to write the equation of the line in point-slope form. y5 x y5 x Step Three: Rewrite the equation in standard form. 5 y 15 x 4 y x x y 19 Ex: Write the equation of the line that is perpendicular to the line x 4y 8that passes through the point, 4 in point-slope form. Step One: Determine the slope of the given line. Step Two: Determine the slope of the new line. 4y x8 y x 4 4 m m 4 Note: The lines are perpendicular, therefore the new line has a slope that is the opposite reciprocal of the slope of the given line. Step Three: Use the slope and the given point to write the equation of the line in point-slope form. 4 4 y x y x 4 4 Page 16 of 0

17 You Try: A line passes through the points 5, 7 and 1, 7. Write the equation of the line. QOD: Compare and contrast the graphs and equations of a set of parallel lines. Sample CCSD Common Exam Practice Question(s): 1. Write an equation for the line that passes through 1, and is parallel to the line whose equation is 4x y. A. y x 1 B. y 4x 6 C. y 4x D. y 4x 9. Which equation describes the pattern in the table? x y A. y 8x B. y 8x C. y x 8 D. y x 8. Which line is perpendicular to the line x y 4? A. y x B. y x C. 1 y x D. 1 y x Page 17 of 0

18 Sample SAT Question(s): Taken from College Board online practice problems. 1. In the xy-coordinate plane above, line l contains the points 0,0 and 1,. If line m (not shown) contains the point 0,0 and is perpendicular to l, what is an equation of m? 1 (A) y x 1 (B) y x 1 (C) y x (D) y 7x (E) y x. The amount that a plumber charges for a service call that is h hours long is shown in the graph above. The charges for a service call consist of an initial amount plus a charge for each hour of work. According to the graph, what is the hourly charge? (A) $5.00 (B) $7.00 (C) $40.00 (D) $50.00 (E) $75.00 Page 18 of 0

19 Syllabus Objective.1 The student will solve application problems using linear models and applying direct variation. Direct Variation: two variables, x and y, vary directly if, for some nonzero constant k, y kx. Note: k is called the constant of variation. The graph of a direct variation equation will pass y through the origin. The equation can also be rewritten as k. x Ex: x and y vary directly. If x when y 15, write the equation that relates the variables. y kx Step One: Find k by substituting x and y into the direct variation equation. 15 k Step Two: Substitute k into the direct variation equation. y 5x k 5 Identifying Direct Variation: given a set of data, use the equation directly. k y to determine if the variables vary x Ex: Determine if the data show direct variation. If yes, write an equation relating the variables. t d Step One: Check whether the ratio of d and t is constant Step Two: If yes, this is the value of k. Substitute this value into the direct variation equation. Solution: d 18t You Try: The income of a store varies directly as the advertising budget. When they spent $000 per month on advertising their income was $10,000. If the owners increase their advertising to $5000 per month, what do they expect their income to be? QOD: How can you determine from a graph of a line if it shows direct variation? Page 19 of 0

20 Sample CCSD Common Exam Practice Question(s): The value of y varies directly with x, and y = 15 when x = 9. What is the value of x when y = 0? A. 15 B. 14 C. 1 D. 9 Sample SAT Question(s): Taken from College Board online practice problems. h If y, where h is a constant, and if y when x 4, what is the value of y equal when x 6? x Grid-In Page 0 of 0

21 Syllabus Objective.5 The student will write the equation of a line or linear function given a set of data. Scatter Plot: a graph of a set of ordered pairs used to determine if there is a relationship between the variables. Note: A scatter plot can show that the data have a positive, negative, or no linear correlation. Correlation: Positive Negative No Linear Correlation Best-Fitting Line (also called the Line of Best Fit): the line that represents the data by following the pattern of the ordered pairs graphed in the scatter plot. Note: The best-fitting line will have a positive slope if the data show positive correlation and a negative slope if the data show negative correlation. Writing the Equation of a Best-Fitting Line: Ex: Draw a scatter plot for the table of values. Determine what type of correlation the data have, and write the equation of the best-fitting line. Tadpoles: Age vs. Tail Length Age (days) Tail (mm) Step One: Graph the ordered pairs in the table using days as the independent variable (horizontal axis) and tail length as the dependent variable (vertical axis). Let d = age in days, and m = length of tail in mm. This data have negative correlation. Step Two: Sketch the best-fitting line. Try to draw the line so that there are as many points above the line as there are below the line. Step Three: Choose two points on the line. Use these two points to write the equation of the line.,15 and 5, m 5 d 15 4 td 45 4t8d 4t5 d 4 t 5 Note: Equations of best-fitting lines may differ slightly. Page 1 of 0

22 You Try: Draw a scatter plot for the Women s Olympic 100 meter times below. Find the equation of a line that closely fits the data. Determine any correlation. Year (19_) y Time (sec) t QOD: The points of a scatter plot lie on a horizontal line. What type of correlation does this show? Explain your answer. Sample SAT Question(s): Taken from College Board online practice problems. The scatterplot above shows the enrollment in 1980 against the enrollment in 000 for twenty colleges. Which of the labeled points represents the college that had the smallest percent increase in enrollment from 1980 to 000? (A) A (B) B (C) C (D) D (E) E Page of 0

23 Syllabus Objective.10 The student will sketch the graphs of linear and absolute value equations and inequalities. Linear Inequality: an inequality that can be written in one of the following forms Ax By C Ax By C Ax By C Ax By C Solution of a Linear Inequality: an ordered pair x, y that makes the inequality true Half-Plane: one of the two regions formed by a line graphed in the coordinate plane Graphing a Linear Inequality: Ex: Graph the linear inequality x y 6. Step One: Graph the line xy 6 Step Two: Determine if the line will be solid or dashed. Dashed **Note: For < and > use a dashed line, and for and use a solid line. Step Three: Choose a test point in either half-plane and substitute it into the original inequality. If it makes the inequality true, then the solutions lie in the same half-plane as the test point. If it does not make the inequality true, then the solutions lie in the other half-plane. Test Point: 0, is not true, so the solutions lie in the other half-plane than the half-plane in which the point 0,0lies. Step Four: Shade the appropriate half-plane. Caution: When choosing a test point, do NOT choose a point ON the line. This represents where the inequality is equal to. You Try: Graph the linear inequality y x. Page of 0

24 Graphing a Linear Inequality in One Variable: Ex: Graph the linear inequality x 4 in the coordinate plane. Step One: Graph the line x 4. Step Two: Determine if the line will be solid or dashed. Solid Step Three: Choose a test point and substitute into the original linear inequality to determine which half-plane to shade. Test Point: 0,0 0 4 is true, so we will shade the half-plane that Step Four: Shade the appropriate half-plane. 0,0 is in. You Try: Graph the linear inequality y in the coordinate plane. QOD: Explain how to determine which half-plane to shade and whether to draw a solid or dashed line when graphing a linear inequality. Page 4 of 0

25 Syllabus Objective.1 The student will define and graph piecewise functions including the greatest integer function. Piecewise Function: a function that is represented by a combination of equations, each corresponding to a part of the domain Evaluating a Piecewise Function: Ex: Find f and f for the function f x x 1, x 0. 4x1 x 0 Step One: Determine which equation to use based upon the value of x. f : x 0, so we will use the first equation. f : x 0, so we will use the second equation. Step Two: Substitute the value of x into the appropriate equation. f f Graphing a Piecewise Function: Ex: Graph the function f x x, x1 x 5 x 1 Step One: Graph the function y x for all values of x less than 1. There should be an open circle on the point 1,1. Step Two: Graph the function y x 5 for all values of x greater than 1 in the same coordinate plane. There should be a closed circle on the point 1,. The Greatest Integer Function: gx x For every real number x, gx is the greatest integer less than or equal to x. The greatest integer function is an example of a step function. Note: All points seen on the graph to the right are solid points. The left endpoints of each segment are open circles. Page 5 of 0

26 You Try: Write the equation of the piecewise function shown in the graph. QOD: In a piecewise function, why must one part of the graph have an open circle as an endpoint, and the other have a closed circle as an endpoint? Sample CCSD Common Exam Practice Question(s): 1. Which graph represents the piecewise function below? x, if x f( x) 5, if x x, if x Page 6 of 0

27 . Which function represents the graph below? A. B. C. D. x1, if 0 x4 f( x) x, if 0 x4 x, if x 1,,,4 f( x) x 1, if x 0, 1,, 1, if 0 x 1, if 1 x f( x), if x 4, if x 4 1, if 0 x 1, if 1 x f( x), if x 4, if x 4 Page 7 of 0

28 Syllabus Objective.10 The student will sketch the graphs of linear and absolute value equations and inequalities. Graphing the Absolute Value Function x x 0 Recall: f x x x x 0 Note: The vertex is at the origin. vertex General Form of the Equation of an Absolute Value Function: y axh k Activity: Discover the effects the variables ah,, and khave on the graph: Plot points or use your graphing calculator to graph the following. Make notes on how the graph changes from the graph of y x. 1. y x. y x. y 1 x 4. y x 5. y x 6. y x1 4 Graphing from the General Form y axh k Vertex: hk, Line of Symmetry: x h If a 0, the V-shaped graph opens up. If a 0, the V-shaped graph opens down. The graph is wider than y x when a 1, and narrower than y x when a 1. Ex: Graph the absolute value function y x1. Step One: Plot the vertex 1,. Step Two: Plot a couple of points to the right of the vertex 0, 1 and 1,1. Step Three: Use symmetry to plot the reflection of the points to the left of the vertex. Page 8 of 0

29 Ex: Write the equation of the graph shown. Step One: Find the vertex and substitute the coordinates in for, 0, hk, hkin the general form. y a x0 Step Two: To find a, choose another point on the graph. Substitute the coordinates into the general equation and solve for a. Point:,1 1 a a a 1 Solution: y x You Try: Graph the absolute value function 1 y x 4. QOD: How does the value of a relate to the slopes of the pieces of an absolute value function to the left and right of the vertex? Sample SAT Question(s): Taken from College Board online practice problems. 1. In the xy-coordinate plane, how many points are a distance of 4 units from the origin? (A) One (B) Two (C) Three (D) Four (E) More than four. 4x 7 5 8x 1 What value of x satisfies both of the equation? Grid-In Page 9 of 0

30 Sample CCSD Common Exam Practice Question(s): Which is the graph of y x 4? A. B. C. D. Page 0 of 0