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x coordinates y coordinates Compare all the x coordinates, repeats. The set is not a function, just a relation. Compare all the x coordinates, no repeats. The set is a function. Compare all the x coordinates in the domain, only one corresponding arrow on each x coordinate. The set is a function. Compare all the x coordinates in the domain, 8 has two corresponding arrows. Repeats The set is not a function, just a relation. When determining if a graph is a function, we will use the Vertical Line Test. Use your pencil as a Vertical Line and place it at the left side of the graph. Slide the pencil to the right and see if it touches the graph ONLY ONCE. If it does it is a FUNCTION. FUNCTION. Use your pencil as a Vertical Line and place it at the left side of the graph. The Vertical Line crosses the graph in 2 or more locations, therefore this graph is just a RELATION. Not multiplication! y coordinates input output y = f(x) Independent Variable Dependent Variable y = 3(4) + 7 y = 12 + 7 y = 19 The work is the same! f(4) = 3(4) + 7 f(4) = 12 + 7 f(4) = 19 Put ( )’s around every x. g x x 5x 3 g 6 6 5 6 3 2 g 6 36 30 3 g 6 3 Substitute -610 for every x. Simplify by Order of Operations. g x x 5x 3 2 g 3t 3t 53t 3 2 g 3t 9t 15t 3 2 g x x 5x 3 2 2 g 10 10 510 3 2 g 10 100 50 3 g 10 147 2 g x x 5x 3 g b 2 b 2 5b 2 3 2 FOIL and distribute b 2b 2 g b 2 b 2 4b 4 5b 10 3 2 b 2b Like 2b Terms, 4 CLT. Combine 2 2 g b 2 b b 4b 4 9b 11 Remember h(x) = y ??? h(x) = y h(3) = 2 =y Find the point when x = 3 j(3) = 5 h(x) = y h(2) = 1 h(x) = y h(0) is not possible! Zero is not in the Domain. h(0) = undefined =y Find the point when x = -2 j(-2) = 1 (3, 5) =y Find the point when x = 0 j(0) = -1 (-2, 1) 0 -2 3 (0, -1) Every x coordinate from 3 to 6 (3, 5) =y Find the point when y = 3 x=2 (6, 5) 5 Find the point when y = 1 3 (2, 3) (-4, 1) 1 x = -4, -2 & 1 (-2, 1) (1, 1) -3 (-?, -3) Find the point when y = 5 Find the point when y = -3 3<x<6 j(x) = -3 is not possible! -3 is not in the Range. [3, 6] interval notation (?, 3) Domain Find the smallest x coordinate to the largest x coordinate. 7 5 -7 Domain: -7 < x < 6 or [-7, 6] 3 6 Range Find the smallest y coordinate to the largest y coordinate. The first set of y coordinates are -4 -4 < y < 3 or (-4, 3). Notice that we started and ended at open circles. The second set of y coordinates are 5 < y < 7 or [5, 7] Open circles mean that the point doesn’t exist and the closed circle means that the point is there. x = -3 at this location…as long as we can touch the graph the x coordinates are there and continuous. Range: -4 < y < 3 or 5 < y < 7 (-4, 3) U [5, 7] Domain Find the smallest x coordinate to the largest x coordinate. Domain: x > -4 or [-4, oo) Range Find the smallest y coordinate to the largest y coordinate. -4 Range: y > -7 or [-7, oo) -7 Domain Find the smallest x coordinate to the largest x coordinate. 4 Domain: -8 < x < 8 or [-8, 8] 1 -8 -1 8 Range Find the smallest y coordinate to the largest y coordinate. The y coordinates are not connected or consistent, therefore we list them separately. Range: {-1, 1, 4} When given the function in set notation, list the x and y coordinates separately. Domain: {-1, 1, 2, 3, 4, 5} Range: {1, 2, 3, 4, 7, 8} Find the domain of the functions. When finding the domain of functions in equation form we will ask ourselves the following questions…. Will the function work when the x is a negative?, …. a zero?, … a positive? If the answers are 3 yes’s, then the domain is all real numbers. If there is a no, then there is a domain restriction we need to find. f x 4x 2 Can I multiply 4 by a negative?, a zero?, a positive? … and then add 2 to the product? ALL Yes! Domain is ALL REAL NUMBERS g x x 2 2 h x x 4 x 3 1 , Can I square a negative?, a zero?, a positive? … and then add 2 to the value? ALL Yes! Domain is ALL REAL NUMBERS , If I square a negative?, a zero?, a positive? … I should be able to raise them to any power! ALL Yes! Domain is ALL REAL NUMBERS , Find the domain of the functions. x2 j x x 1 k x x 9 Adding and subtracting always is a Yes…Can I divide by a negative?, a zero?, a positive? NO! Can’t divide by ZERO! Set the denominator equal to zero and solve for x to find the restriction. Domain is ALL REAL NUMBERS, except 1 x 1 Can I take the absolute value of a negative?, a zero?, a positive? ALL Yes! Domain is ALL REAL NUMBERS , 5 m x 2 x 9 We have a fraction again, set the bottom equal to zero and solve for x. x2 9 0 x 3x 3 0 x 3 Domain is ALL REAL NUMBERS, except for -3 and 3. ,3 3,3 3, 8 qx 2 x 6 x 16 We have a fraction again, set the bottom equal to zero and solve for x by factoring. x 2 6 x 16 0 x 8x 2 0 x 8,2 Domain is ALL REAL NUMBERS, except for -8 and 2. ,8 8,2 2, 2 p x 3 x x 2 12 x We have a fraction again, set the bottom equal to zero and solve for x by factoring. 3 2 x x 12 x 0 x x 2 x 12 xx 4x 3 0 x 0,3,4 Domain is ALL REAL NUMBERS, except for -3, 0 and 4. ,3 3,0 0,4 4, Domain Restrictions 5 < -5, FALSE -5 < 5 < 3 , FALSE 5 > 3, TRUE x = 5, test it in the domain restrictions to see which one is true! Substitute the 5 into that function. f 5 5 9 4 f 5 4 x = -7, and -7 < -5. Substitute -7 into the first function. f 7 3 7 2 f 7 23 x = -5, and -5 < -5 < 3. Substitute -5 into the second function. f 5 5 1 2 f 5 24 x = 3, and 3 > 3. Substitute 3 into the third function. f 3 3 9 6 f 3 6 y f ( x) c f ( x) 3 y f ( x) x 2 y f ( x) x y f ( x) x1 y f ( x) mx b y f ( x) x n x n1 c y f ( x) x 3 Cubic Func. y f ( x) 1 x (0, 6) rise m = slope = run b = y-int = (0, b) starting point y-int = (0, 6) directions -5 m= 2 down 5 right 2 down 5 right 2 up 1 right 3 right 3 up 1 point = (x1, y1) m = slope = rise run starting point y-int = (-3, 4) directions 1 m= 3 Or in reverse left 3 up 1 down 1 (-3, 4) right 3 A, B, and C are integers. To graph find x and y intercepts ??? To find the y intercept the x coordinate is zero! (0, y) 20 3 y 18 3 y 18 y 6 0,6 To find the x intercept the y coordinate is zero! ( x, 0) 2x 30 18 2x 18 x 9 9,0 Doesn’t fit, but that is ok…we can use the slope! A 2 2 m B 3 3 Notice that there is no y variable in the equation. This means we can’t cross the y axis! Must be a VERTICAL LINE at x = 6 rise m = slope = 0 = undefined Notice that there is no x variable in the equation. This means we can’t cross the x axis! Must be a HORIZONTAL LINE at y = - 4 0 m = slope = run = 0 To graph find x and y intercepts. We can see that 3 will divide into -9 evenly, but 5 won’t. So we should find the x intercept and the slope to graph this line. To find the x intercept the y coordinate is zero! ( x, 0) 3x 50 9 3x 9 x 3 Find the slope! A 3 m B 5 3,0 Write the equation of a line that contains the points ( 3, 8 ) and ( 5, -1 ). Find the slope first. m y2 y1 1 8 9 x2 x1 53 2 Next use the point-slope form to write the equation. y y1 mx x1 9 9 y 8 x 3 Convert to y = mx + b. y x 3 8 2 2 9 27 9 43 y x 8 x 2 2 2 2 Yellow TAXI Cab Co. charges a $10 pick-up fee and charges $1.25 for each mile. Write a cost function, C(m) that is dependent on the miles, m, driven. Remember…functions are equal to y. y = C(m). Use y = mx + b. The slope is the same as rate! The y intercept (b) is the starting point or initial cost. The $10 pick-up fee is a one time charge or initial cost. b = 10 The $1.25 for each mile is a rate. m = 1.25 Replace y with C(m) and x with m. y 1.25 x 10 Cm 1.25m 10 In the year 2000, the life expectancy of females was 83.5. In 2004, it was 86.5. Write a linear function E(t) where t is the number of years after 2000 and E(t) is the life expectancy in t years. Estimate the life expectancy in the year 2009. Estimate the when the life expectancy will be 94. Looks difficult only because of all the words! Understand the data given to write the equation of a line! This looks like points (x, y) = (t, E(t)) Year # of years after 2000 (t) Age E(t) (0, 83.5) 2000 0 83.5 2004 4 86.5 (4, 86.5) We are back to the first problem we did for writing the equation of a line. Use y = mx + b because we are working with functions and (0, 83.5) is the y intercept….b is 83.5. 86.5 83.5 3 Find the slope between the points. m 0.75 40 4 y 0.75 x 83.5 Estimate the life expectancy in the year 2009. t 9 Et 0.75t 83.5 E9 0.759 83.5 90.25 Estimate the when the life expectancy will be 94. Et 94 94 0.75t 83.5 10.5 0.75t t 14 83.5 83.5 0.75 0.75 14 years past the year 2000, 2014. In the year 2003, a certain college had 3450 students. In the year 2008, the college had 4100 students. Write a linear function P(t) where t is the number of years after 2000 and P(t) is the population of the college. Estimate the population in the year 2012. Estimate the year when the population will reach 5400. Understand the data given to write the equation of a line! Year 2003 2008 # of years after 2000 (t) 3 8 Students P(t) 3450 4100 Points (x, y) = (t, P(t)) (3, 3450) (8, 4100) Use y = mx + b because we are working with functions, but this time we will have to solve for b. 4100 3450 650 130 Find the slope between the points. m 83 5 y 130 x b Plug in a point, (8, 4100). Estimate the population in the year 2012. 4100 1308 b y 130 x 3060 t 12 P12 13012 3060 4620 4100 1040 b Pt 130t 3060 3060 b Estimate the year when the population will reach 5400. Pt 5400 5400 130t 3060 18 t 2340 130t 18 years past the year 2000 is the year 2018. Same Line 4 7 3, true 5 4 7 27 20 7 27, true Yes, (-4, 7) is a solution. Find the solution to the system by graphing. x y 1 1 m 1 1 x y 3 1 m 1 1 ( 1, 2 ) Solution is ___________. Find the solution to the system by graphing. y 3x 2 Convert to y = mx + b 6 x 2 y 12 6x 6x 2 y 6 x 12 2 2 2 y 3x 6 The slopes are the same and the y-intercepts are different. Solution is ___________. No Solution Find the solution to the system by graphing. 2 x 3 y 6 Divide everything by 4. 8 x 12 y 24 4 4 4 2 x 3 y 6 Same LINE! Infinite Solutions, but not the final answer. Convert to y = mx + b 3 y 2 x 6 2 y x2 3 2 x, x 2 Solution is ___________. 3 Answer must be written as a point. Section 8.2. Solving linear systems by SUBSTITUTION & ELIMINATION. Solve for x & y. 3x 2 y 11 x y 3 Substitution Method. 1. Choose an equation and get x or y Step 1 by itself. 2. Substitute step 1 equation into the second equation. x y 3 y x3 Step 2 3x 2 y 11 3 x 2 x 3 11 Step 3 3 x 2 x 6 11 3. Solve for the remaining variable. 4 Substitute this answer into the step 1 Step 4 y x 3 equation. y 1 3 y4 5 x 6 11 5x 5 x 1 (1,4) Is the intersection point and solution. Solve the system for x and y. x 3y 1 y 2x 3 x 2y 5 2 x 4 y 11 x 32x 3 1 22 y 5 4 y 11 x 6x 9 1 4 y 10 4 y 11 5x 9 1 5x 10 x 2 y 2x 3 y 2 2 3 y 4 3 1 2,1 10 11 False Stmt. No Solution 2 x2 3 10 x 15 y 30 y 2 10 x 15 x 2 30 3 10x 10x 30 30 30 30 True Stmt. Infinite Solutions But not done! Answer should be a point ( x, mx + b ) 2 x2 3 2 x, x 2 3 y Sect 8.1 Systems of Linear Equations Solve for x & y. 3x 4 y 1 2 x 3 y 12 Elimination Method. 1. Choose variable to cancel out. Look for opposite signs. 2. Add the equations together to cancel. 3. Solve for the remaining variable. 4 Substitute this answer into either equation in the step 1 equations. Step 4 Step 1 The y-terms are opposite signs. Multiply the first equation by 3 and the second equation by 4. 3x 4 y 13 9 x 12 y 3 Step 2 + 2 x 3 y 124 8 x 12 y 48 23 3 y 12 6 3 y 12 3y 6 y2 17 x 0 y 51 17 x 51 Step 3 17 17 x3 (3, 2) is the intersection point and solution. Solve the system for x and y. The y-terms are 2 5 x 4 y 22 2 5 x 7 y 29 and easiest to cancel. To multiples of 4. Multiply determine which factor will 3x 8 y 18 the 1st equation by -2 to 5 2 x 3 y 17 be negative check the y-terms + make opposites. 10 x 8 y 44 10 x 14 y 58 Add equations together. + 10 x 15 y 85 13x 26 y 27 x2 The x-terms are the smallest 52 4 y 22 10 4 y 22 4 y 12 y 3 2,3 2x 327 17 2x 81 17 2 x 64 x 32 32,27 Solve the system for x and y. Remove fractions by 2 3 35 x y 5 multiply by the LCD and decimals by multiply by 10’s 7 5 10 0.8 x 0.3 y 4.1 10 x 21 y 175 8 x 3 y 41 7 + 10 x 21 y 175 56 x 21 y 287 66 x 462 x7 107 21y 175 70 21y 175 21 y 105 y 5 7,5 2 5 x 8 y 11 10x 16 y 22 + 5 +2 x 8 y 13 10x 40 y 65 7x 24 56 y 43 24 43 x y 7 56 24 2 8 y 13 7 48 8 y 13 7 43 1 1 8y 7 8 8 43 y 56 24 43 , 7 56 Solve the system for x and y. 2 3x 2 y 4 6x 4 y 6 6x 4 y 8 0 14 False Stmt. No Solution 8 x 2 y 4 2 4x y 2 4 x y 2 00 True Stmt. Infinite Solutions Solve for y. 4 x y 2 4x 2 y Answer should be a point ( x, mx + b ) x,4x 2 Total-Relationship Systems. In 2008, there were 746 species of plants that were considered threatened or endangered. The number considered threatened was 4 less than a fourth of the number considered endangered. How many plants are considered threatened and endangered in 2008? Find the two unknown’s from the question and determine their TOTAL. Always read the question sentence first. 1 T + E = 746 4 E 4 E 746 T = How many threatened? 146 1.25E 4 746 E = How many endangered? 600 Now read through the details to 1.25E 750 T = ¼( E ) – 4 find the RELATIONSHIP between 1.25 1.25 the two variables. Substitution Method E 600 T E 746 T 600 746 T 146 Total-Relationship System. The sum is 90 o Two angles are complementary. One angle is 12o less than twice the other. Find the measure of the two angles. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. 2B 12 B 90 A + B = 90 A = First angle 56 3B 12 90 B = Second angle 34 3B 102 A = 2( B ) – 12 Now read through the details to 3 3 find the RELATIONSHIP between B 34 Substitution Method the two variables. A B 90 A 34 90 A 56 *** If the two angles are supplementary, then the sum is 180 o Total-Rate Systems. A jewelry designer purchased 80 beads for a total of $39. Some of the beads were silver beads that cost 40 cents each and the rest were gold beads that cost 65 cents each. How many of each type did the designer buy? Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. 0.40G 0.40S 32 0.40 G + S = 80 G = How many Gold beads? 28 0.65G 0.40 S 39 S = How many Silver beads? 52 0.25G 7 Now read through the details to 0.65G + 0.40S = 39.00 0.25 0.25 find the RATE on each variable. Multiply the rates to the variables Elimination Method G 28 Cancel smallest variable term. and set equal to total cost. G S 80 28 S 80 S 52 Total-Rate Systems. Jane’s student loans total $9,600. She has a PLUS loan at 8.5% and a Stafford loan at 6.8% simple interest. In one year, she was charged $729.30 in simple interest. How much was each loan? Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. 0.068 P + S = 9600 P = How much was the PLUS loan? $4500 S = How much was the Stafford loan? $5100 0.068P 0.068S 652.80 0.085P 0.068S 729.30 Now read through the details to 0.085P + 0.068S = 729.30 find the RATE on each variable. Multiply the rates to the variables Elimination Method and set equal to the total cost. Cancel smallest variable term. Percent must be changed to a decimal! 0.017P 76.50 0.017 0.017 P 4500 P S 9600 4500 S 9600 S 5100 Total-Rate Systems. A child ticket costs $3 and an adult ticket costs $5 at an afternoon movie. 300 tickets were sold for $1,150. How many of each type of ticket were purchased? Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. 3C 3 A 900 3 C + A = 300 C = How many Child tickets? 175 A = How many Adult tickets? 125 3C + 5A = 1150 Now read through the details to find the RATE of each variable. Multiply the rates to the variable Elimination Method and set equal to total cost. Cancel smallest variable term. 3C 5 A 1150 2 A 250 2 2 A 125 A C 300 125 C 300 C 175 Total-Mixture Systems. Cashews cost $4 per pound and Walnuts cost $10 per pound. How much of each type should be used to make a 50 pound mixture that sells for $5.80 per pound? Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. C = How many pounds of Cashews? 35 W = How many pounds of Walnuts? 15 4 C + W = 50 4C 4W 200 4C 10W 290 Now read through the details to 4C + 10W = 5.80(50) find the RATE of each variable and TOTAL. Multiply the rates to the Elimination Method variable and TOTAL. Cancel smallest variable term. 6W 90 6 6 W 15 C W 50 C 15 50 C 35 Total-Mixture Systems. Home Depot carries two brands of liquid fertilizers containing nitrogen and water. Gentle Grow is 3% nitrogen while Super Grow is 8% nitrogen. Home Depot needs to combine the two types of solution to fill a customer’s order that requested 90L of fertilizer that is 6% nitrogen. How much of each brand should be used to fill the order? Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. G = How many liters of Gentle Grow? 0.03 G + S = 90 36 S = How many liters of Super Grow? 54 0.03G 0.03S 2.7 0.03G 0.08S 5.4 0.03G + 0.08S = 0.06(90) Now read through the details to find the RATE of each variable and Elimination Method TOTAL. Multiply the rates to the Cancel smallest variable term. variable and TOTAL. 0.05S 2.7 0.05 0.05 S 54 G S 90 G 54 90 G 36 Distance Systems. A jet flies 4 hours west with a 60 mph tailwind. Returning against the same wind, the jet takes 5 hours. What is the speed of the jet with no wind? D With wind D Against wind D = r * r 60 r 60 t 4 D 4r 240 Substitution Method 5 D 5r 300 4r 240 5r 300 4r 4r 240 r 300 300 300 540 r 540 mph in no wind. Distance Systems. A freight train leaves Chicago heading to Denver at a speed of 40 mph. Two hours later an Amtrak train leaves Chicago bound for Denver at a speed of 60 mph. How far will the trains travel until the Amtrak passes the freight train? D = r * t Freight train D 40 t Amtrak D 60 t 2 D 40t Substitution Method D 60t 120 D 40t 40t 60t 120 60t 60t D 406 20t 120 D 240 20 20 t 6 They will travel 240 miles.