Introdução a Função

23 Graph Linear Inequalities Student Outcomes Students recognize and identify solutions to twovariable inequalities They represent the solution set graphically They create twovariable inequalities to represent a situation Students understand that a halfplane bounded by the line 𝑎𝑥 𝑏𝑦 𝑐 is a visual representation of the solution set to a linear inequality such as 𝑎𝑥 𝑏𝑦 𝑐 They interpret the inequality symbol correctly to determine which portion of the coordinate plane is shaded to represent the solution Sumary An ordered pair is a solution to a twovariable inequality if when each number is substituted into its corresponding variable it makes the inequality a true number sentence Each ordered pair of numbers in the solution set of the inequality corresponds to a point on the coordinate plane The set of all such points in the coordinate plane is called the graph of the inequality The graph of a linear inequality in the coordinate plane is called a halfplane Exercises The solution to 𝑥 𝑦 20 is shown on the graph below a Graph the solution to 𝑥 𝑦 20 b Graph the solution to 𝑥 𝑦 20 c Graph the solution to 𝑥 𝑦 20 d Graph the solution to 𝑥 𝑦 20 2 Match each inequality with its graph Explain your reasoning a 2𝑥 𝑦 6 b 𝑦 2𝑥 6 c 2𝑥 𝑦 6 d 2𝑥 6 𝑦 3 Graph the solution set in the coordinate plane Support your answer by selecting two ordered pairs in the solution set and verifying that they make the inequality true a 15𝑥 𝑦 30 b 5 𝑦 c 𝑦 75𝑥 16 d 2𝑥 8𝑦 25 e 4𝑥 𝑦 f 3𝑥 0 4 Marti sells tacos and burritos from a food truck at the farmers market She sells burritos for 450 each and tacos for 300 each She hopes to earn at least 150 at the farmers market this Saturday a Identify three combinations of tacos and burritos that will earn Marti more than 150 b Identify three combinations of tacos and burritos that will earn Marti exactly 150 c Identify three combinations of tacos and burritos that will not earn Marti at least 150 d Graph your answers to parts ac in the coordinate plane and then shade a halfplane that contains all possible solutions to this problem e Create a linear inequality that represents the solution to this problem Let 𝑥 equal the number of burritos that Marti sells and let 𝑦 equal the number of tacos that Marti sells f Is the point 10 495 a solution to the inequality you created in part e Explain your reasoning 21 Graph Linear Equations Student Outcomes Students understand that an equation is a statement of equality between two expressions When values are substituted for the variables in an equation the equation is either true or false Students find values to assign to the variables in equations that make the equations true statements Exercises 1 In the following exercises plot each point in a rectangular coordinate system A 20 B 04 C 05 D 30 2 In the following exercises determine which ordered pairs are solutions to the given equations 21 𝑦 6𝑥 2 A 14 B 1 3 0 C 6 2 3 Graph a Linear Equation by Plotting Points In the following exercises graph by plotting points A 𝑦 3𝑥 B 𝑦 4 5 𝑥 1 C 2𝑥 𝑦 7 4 Graph Vertical and Horizontal lines In the following exercises graph each equation 𝑦 2 5 In the following exercises graph each pair of equations in the same rectangular coordinate system 𝑦 2𝑥𝑦 2𝑥 𝑎𝑛𝑑 𝑦 2 6 Find x and yIntercepts In the following exercises find the x and yintercepts A 7 In the following exercises find the intercepts of each equation A 𝑥 𝑦 1𝑥 𝑦 1 B 2𝑥 3𝑦 122𝑥 3𝑦 12 C 𝑦 3𝑥 8 Graph a Line Using the Intercepts In the following exercises graph using the intercepts A 𝑥 𝑦 4𝑥 𝑦 4 B 2𝑥 4𝑦 82𝑥 4𝑦 8 Find the Slope of a Line In the following exercises find the slope of each line shown A 9 25 Functions Student Outcomes Students use function notation evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context Students create functions that represent a geometric situation and relate the domain of a function to its graph and to the relationship it describes Exercises Find the indicated functional values 1 If 𝑓𝑥 𝑥2 1 𝑓5 𝑓3 𝑓3 y 𝑓6 2 If 𝑓𝑥 2𝑥2 𝑥 𝑓5 𝑓 1 2 𝑓2 y 𝑓7 3 If 𝑓𝑥 𝑥 1 𝑓1 𝑓0 𝑓3 y 𝑓5 4 If 𝑓𝑥 2𝑥 4 𝑓 1 2 𝑓 1 2 𝑓 5 2 y 𝑓4 5 If 𝑓𝑥 4𝑥 2 6 If 𝑓𝑥 10 1𝑥 7 If 𝑓𝑥 2𝑥5 𝑥𝑥3 Determine if the graph in the figure is the graph of a function 8 10 In problems 1 and 2 use the graph of the given function 𝑓 in the figure to estimate the values 𝑓3 𝑓2 𝑓1 𝑓1 𝑓2 𝑦 𝑓3 if it is possible Calculate the 𝑦 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 12 13 B LATAM Global School 21 Graph linear Equations 1 ys c 05 4 3 2 1 0 1 2 3 4 4 3 2 1 0 1 2 3 4 x A 20 B 04 D 30 1 2 An ordered pair ij is a solution to the linear equation y axb if and only if j aib 21 y 6x 2 A 61 2 6 2 4 14 is solution to the equation B 6 13 2 2 2 0 130 is solution to the equation C 66 2 36 2 34 2 62 is not solution to the equation 2 Linear equations are graphically represented by a line That means we only need two distinct points to draw it A for x0 y 30 0 00 for x1 y 31 3 13 B for x0 y 45 x 1 0 1 1 01 for x5 y 45 5 1 4 1 5 55 c for x0 20 y 7 y 7 07 for y0 2x 0 7 2x 7 x 72 35 350 4 Notice that for every x the value of y remains 2 y2 y 2xy 2x leads to a 3 equations system y 2xy I y 2x II 2xy 2x III I if y0 0 2x0 no The equation is true for any value in x II 0 2x x 02 0 Given I and II the only answer is 00 Verifying if 00 fits for III 200 20 00 If y 0 y 2xy 1 2x x 12 II y 212 1 Given I and II the only answer is 121 Verifying if 121 fits for III 2121 212 11 Therefore y 2xy 2x has 2 solutions 00 and 12 1 y2 is graphically represented by a horizontal line that intercepts the yaxis at y 2 6 The line crosses the xaxis at x 4 and y 0 so xintercept is the point 40 Similarly the line crosses the yaxis at x 0 and y 4 Thus yintercept is the point 04 7 Algebraically we can find the xintercept by finding the points which y0 and the yintercept by finding the points which x 0 A for x0 0y 10 y 1 y 1 y 1 y intercept is 01 for y0 x0 x0 1 x x 1 Notice that the only value that makes x x true is x 0 Therefore there is no x intercept 8 b For x 0 20 3y 12 20 3y 12 3y 12 y 123 4 y intercept is 04 for y 0 2x 30 12 2x 12 The only value that makes 2x 12 true is x 0 Thus there is no x intercept c For x 0 y 30 0 y intercept is 00 Since y0 00 is also x intercept for y 0 0 3x x 03 0 x intercept is 00 Slope A Given 2 distinct points x1 y1 and x2 y2 we can find the slope of the line by the formula m y2 y1 x2 x1 00 and 16 are points of the line m 6 0 1 0 6 1 6 B 4 4 and 1 3 are points of the line m 3 4 1 4 3 4 1 4 1 3 1 f5 52 1 25 1 24 f3 32 1 3 1 2 f3 32 1 9 1 8 f6 62 1 36 1 35 2 f5 252 5 225 5 55 f12 2122 12 214 12 12 12 1 f2 222 2 24 2 6 f7 272 7 249 7 91 3 f1 1 1 0 f0 0 1 1 f3 3 1 2 f5 5 1 6 4 f12 212 4 1 4 3 f12 212 4 1 4 5 f52 252 4 5 4 3 f4 24 4 8 4 12 43 23 5 6 7 there is no value indicated 8 It is not the graph of a function 9 It is the graph of a function 10 It is the graph of a function 11 It is not the graph of a function of one real variable 12 f3 0 f2 035 f1 02 yinterception 00 f1 035 f2 27 f3 45 13 f3 065 f2 93 f1 095 yinterception 0 08 f1 01 f2 08 f3 045 14 f1 22 yintercept 00 f1 11 f2 37 f3 38 15 f2 23 f1 02 yintercept 00 f1 08 a Graph the solution to x y 20 b Graph the solution to x y 20 c Graph the solution to x y 20 d Graph the solution to x y 20 2 a 2 First of all 2x y is strictly greater than 6 ie the equality graphically represented by the line should not be considered Besides that we can test a point that doesnt lie on the line to find which region should be considered For example 00 2 0 0 6 0 6 is false so 00 does not lie on the region b 4 The inequality is not strict and 09 doesnt satisfy the inequality c 3 The inequality is strict and 00 satisfy it d 1 The inequality isnt strict and 00 satisfy it 3 a 30 153 0 30 45 30 031 15 0 31 30 31 30 b 00 5 0 11 5 1 c 00 0 750 16 0 16 20 0 752 16 0 31 d 00 20 80 25 0 25 11 21 81 25 6 25 e 11 41 1 4 1 04 40 4 0 4 f 41 31 0 3 0 37 33 0 9 0 4 t number of sold tacos b number of sold burritos 45b 3t 150 a 1 51 tacos 2 34 burritos 3 50 tacos and 34 burritos b 1 50 tacos 2 2 burritos and 47 tacos 3 4 burritos and 44 tacos c 1 Nothing sold 2 49 tacos 3 33 burritos e 45x 3y 150 f 4510 3495 1935 150 Yes it is a solution because the point satisfies the inequality