1. x coordinates y coordinates Compare all the x coordinates, Compare all the x coordinates, no repeats. The set is a function. The set is not a function, just a relation. repeats.

2. 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.

3. 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.

4. y coordinatesinput output y = f(x) Dependent Variable Independent Variable y = 3(4) + 7 y = 12 + 7 y = 19 f(4) = 3(4) + 7 f(4) = 12 + 7 f(4) = 19 The work is the same! Not multiplication!

5. Substitute -6 for every x. Simplify by Order of Operations. 10 Put ( )’s around every x. FOIL and distribute Combine Like Terms, CLT.

6. Remember h(x) = y h(x) = y h(3) = 2 h(x) = y h(2) = 1 h(x) = y ??? = y Find the point when x = 3 3 (3, 5) j(3) = 5 (-2, 1) = y Find the point when x = -2 -2 j(-2) = 1 (0, -1) 0 = y Find the point when x = 0 j(0) = -1 h(0) is not possible! Zero is not in the Domain. h(0) = undefined

7. 3 (2, 3) = y Find the point when y = 3 x = 2 (1, 1) Find the point when y = 1 x = -4, -2 & 1 (-2, 1) 5 (6, 5)(3, 5) Find the point when y = 5 3 < x < 6 [ 3, 6 ] interval notation (?, 3)(-?, -3) -3 1 Find the point when y = -3 j(x) = -3 is not possible! -3 is not in the Range. Every x coordinate from 3 to 6 (-4, 1)

8. -7 Domain Find the smallest x coordinate to the largest x coordinate. 6 Domain: -7 < x < 6 or [ -7, 6 ] -4 3 Range Find the smallest y coordinate to the largest y coordinate. 5 7 The first set of y coordinates are -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 ] Range: -4 < y < 3 or 5 < y < 7 (-4, 3) U [ 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.

9. -4 Domain Find the smallest x coordinate to the largest x coordinate. Domain: x > -4 or [ -4, oo ) -7 Range Find the smallest y coordinate to the largest y coordinate. Range: y > -7 or [ -7, oo )

10. -8 Domain Find the smallest x coordinate to the largest x coordinate. Domain: -8 < x < 8 or [ -8, 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 } 8 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 }

11. 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. 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 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

12. Find the domain of the functions. 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 Can I take the absolute value of a negative?, a zero?, a positive? ALL Yes!Domain is ALL REAL NUMBERS We have a fraction again, set the bottom equal to zero and solve for x. Domain is ALL REAL NUMBERS, except for -3 and 3.

13. We have a fraction again, set the bottom equal to zero and solve for x by factoring. Domain is ALL REAL NUMBERS, except for -8 and 2. We have a fraction again, set the bottom equal to zero and solve for x by factoring. Domain is ALL REAL NUMBERS, except for -3, 0 and 4.

14. Domain Restrictions x = 5, test it in the domain restrictions to see which one is true! Substitute the 5 into that function. 5 < -5, FALSE -5 < 5 < 3, FALSE 5 > 3, TRUE x = -7, and -7 < -5. Substitute -7 into the first function. x = -5, and -5 < -5 < 3. Substitute -5 into the second function. x = 3, and 3 > 3. Substitute 3 into the third function.

15. Cubic Func.

16. m = slope = -5 2 b = y-int = (0, b) y-int = (0, 6) m = (0, 6) rise run starting point directions down 5 right 2 down 5 right 2

17. m = slope = rise run point = (x 1, y 1 ) 1313 y-int = (-3, 4) m = starting point directions (-3, 4) up 1 right 3 up 1 right 3 up 1 right 3 down 1 left 3 Or in reverse

18. 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) To find the x intercept the y coordinate is zero! ( x, 0) Doesn’t fit, but that is ok…we can use the slope! ???

19. m = slope = = undefined 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 0 m = slope = = 0 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 run

20. 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) Find the slope!

21. Write the equation of a line that contains the points ( 3, 8 ) and ( 5, -1 ). 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. Find the slope first. Next use the point-slope form to write the equation. Convert to y = mx + b. 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.

22. 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! Year # of years after 2000 (t) Age E(t) 2000 0 83.5 2004 4 86.5 This looks like points (x, y) = (t, E(t)) 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. Find the slope between the points. (0, 83.5) (4, 86.5) Estimate the life expectancy in the year 2009. Estimate the when the life expectancy will be 94. 14 years past the year 2000, 2014.

23. 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 # of years after 2000 (t) Students P(t) 2003 3 3450 2008 8 4100 Points (x, y) = (t, P(t)) Use y = mx + b because we are working with functions, but this time we will have to solve for b. Find the slope between the points. (3, 3450) (8, 4100) Plug in a point, (8, 4100). Estimate the population in the year 2012. Estimate the year when the population will reach 5400. 18 years past the year 2000 is the year 2018.

24. Same Line Yes, (-4, 7) is a solution.

25. Find the solution to the system by graphing. Solution is ___________. ( 1, 2 )

26. Find the solution to the system by graphing. Solution is ___________. No Solution Convert to y = mx + b The slopes are the same and the y-intercepts are different.

27. Find the solution to the system by graphing. Solution is ___________. Divide everything by 4. Same LINE! Infinite Solutions, but not the final answer. Convert to y = mx + b Answer must be written as a point.

28. Substitution Method. 1. Choose an equation and get x or y by itself. 2. Substitute step 1 equation into the second equation. 3. Solve for the remaining variable. 4 Substitute this answer into the step 1 equation. Solve for x & y. Is the intersection point and solution. Step 1Step 2 Step 3 Step 4 Section 8.2. Solving linear systems by SUBSTITUTION & ELIMINATION.

29. Solve the system for x and y. False Stmt. No Solution True Stmt. Infinite Solutions But not done! Answer should be a point ( x, mx + b )

30. Sect 8.1 Systems of Linear Equations 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. Solve for x & y. (3, 2) is the intersection point and solution. The y-terms are opposite signs. Multiply the first equation by 3 and the second equation by 4. Step 1 Step 2 + Step 3 Step 4

31. Solve the system for x and y. The y-terms are multiples of 4. Multiply the 1 st equation by -2 to make opposites. Add equations together. The x-terms are the smallest and easiest to cancel. To determine which factor will be negative check the y-terms + +

32. Solve the system for x and y. Remove fractions by multiply by the LCD and decimals by multiply by 10’s + + +

33. Solve the system for x and y. False Stmt. No Solution True Stmt. Infinite Solutions Solve for y. Answer should be a point ( x, mx + b )

34. 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? Total-Relationship Systems. Find the two unknown’s from the question and determine their TOTAL. Always read the question sentence first. T = How many threatened? E = How many endangered? T + E = 746 Now read through the details to find the RELATIONSHIP between the two variables. T = ¼( E ) – 4 Substitution Method

35. Two angles are complementary. One angle is 12 o less than twice the other. Find the measure of the two angles. Total-Relationship System. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. A = First angle B = Second angle The sum is 90 o A + B = 90 Now read through the details to find the RELATIONSHIP between the two variables. A = 2( B ) – 12 Substitution Method *** If the two angles are supplementary, then the sum is 180 o

36. 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? Total-Rate Systems. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. G = How many Gold beads? S = How many Silver beads? G + S = 80 Now read through the details to find the RATE on each variable. Multiply the rates to the variables and set equal to total cost. 0.65G + 0.40S = 39.00 Elimination Method Cancel smallest variable term.

37. 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? Total-Rate Systems. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. P = How much was the PLUS loan? S = How much was the Stafford loan? P + S = 9600 Now read through the details to find the RATE on each variable. Multiply the rates to the variables and set equal to the total cost. Percent must be changed to a decimal! 0.085P + 0.068S = 729.30 Elimination Method Cancel smallest variable term.

38. 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? Total-Rate Systems. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. C = How many Child tickets? A = How many Adult tickets? C + A = 300 Now read through the details to find the RATE of each variable. Multiply the rates to the variable and set equal to total cost. 3C + 5A = 1150 Elimination Method Cancel smallest variable term.

39. 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? Total-Mixture Systems. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. C = How many pounds of Cashews? W = How many pounds of Walnuts? C + W = 50 Now read through the details to find the RATE of each variable and TOTAL. Multiply the rates to the variable and TOTAL. 4C + 10W = 5.80(50) Elimination Method Cancel smallest variable term.

40. 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? Total-Mixture Systems. Find the two unknown from the question and determine their TOTAL. Always read the question sentence first. G = How many liters of Gentle Grow? S = How many liters of Super Grow? G + S = 90 Now read through the details to find the RATE of each variable and TOTAL. Multiply the rates to the variable and TOTAL. 0.03G + 0.08S = 0.06(90) Elimination Method Cancel smallest variable term.

41. 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? Distance Systems. With wind Against wind D = r * t Substitution Method 540 mph in no wind.

42. 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? Distance Systems. Freight train Amtrak D = r * t Substitution Method They will travel 240 miles.