Tables, Functions & Graphs

1. Tables, Functions & Graphs MA.6.A.3.6, MA.6.A.3.2

2. Sunshine State Standards MA.6.A.3.6 Construct and analyze tables…and equations to describe linear functions…using both common language and algebraic notation. AlsoMA.6.A.3.2

3. Vocabulary function input, x output, y linear linear equation

4. A function is a rule that relates two quantities so that each input value corresponds exactly to one output value.

5. Warm Up Write an equation for each function. Tell what each variable you use represents. 1. The length of a wall is 4 ft more than three times the height. 2. The number of trading cards is 3 less than the number of buttons. l = 3h + 4, where l is length and h is height. c = b – 3, where c is the number of cards and b is the number of buttons.

6. x 0 1 2 3 4 10 y 4 7 10 13 16 Compare x and y to find a pattern. Use the pattern to write an equation. Substitute 10 for x. Use your rule to find y when x = 10. Additional Example 1: Writing Equations from Function Tables Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x. y is 3 times x plus 4. y = 3x + 4 y = 3(10) + 4 y = 30 + 4 = 34

7. x 3 4 5 6 7 10 y 10 12 14 16 18 Compare x and y to find a pattern. Use the pattern to write an equation. Substitute 10 for x. Use your function rule to find y when x = 10. Check It Out: Example 1 Write an equation for a function that gives the values in the table. Use the equation to find the value of y for the indicated value of x. y is 2 times x + 4. y = 2x + 4 y = 2(10) + 4 y = 20 + 4 = 24

8. You can write equations for functions that are described in words.

9. Choose variables for the equation. Write an equation. Additional Example 2: Translating Words into Math Write an equation for the function. Tell what each variable you use represents. The height of a painting is 7 times its width. h = height of painting w = width of painting h = 7w

10. 1 Understand the Problem Additional Example 3: Problem Solving Application The school choir tracked the number of tickets sold and the total amount of money received. They sold each ticket for the same price. They received $80 for 20 tickets, $88 for 22 tickets, and $108 for 27 tickets. Write an equation for the function. The answer will be an equation that describes the relationship between the number of tickets sold and the money received.

11. 2 Make a Plan 3 Solve t 20 22 27 m 80 88 108 Compare t and m. Write an equation. You can make a table to display the data. Let t be the number of tickets. Let m be the amount of money received. m is equal to 4 times t. m = 4t

12. 4 Look Back ? ? ? 80= 4•20 88= 4•22 108= 4•27 ? ? ? 80= 80 88= 88 108= 108 Substitute the t and m values in the table to check that they are solutions of the equation m = 4t. m = 4t (20, 80)‏ m = 4t (22, 88)‏ m = 4t (27, 108)‏

13. Make a function table by using the given values for x to find values for y. Write these solutions as ordered pairs. x 4x + 2 y 1 4(1) + 2 6 2 4(2) + 2 10 3 4(3) + 2 14 4 4(4) + 2 18 Additional Example 1: Finding Solutions of Equations with Two Variables Use the given x-values to write solutions of the equation y = 4x + 2 as ordered pairs. x = 1, 2, 3, 4. (x, y)‏ (1, 6)‏ (2, 10)‏ (3, 14)‏ (4, 18)‏

14. Make a function table by using the given values for x to find values for y. Write these solutions as ordered pairs. x 3x + 2 y 2 3(2) + 2 8 3 3(3) + 2 11 4 3(4) + 2 14 5 3(5) + 2 17 Check It Out: Example 1 Use the given x-values to write solutions of the equation y = 3x + 2 as ordered pairs. x = 2, 3, 4, 5. (x, y)‏ (2, 8)‏ (3, 11)‏ (4, 14)‏ (5, 17)‏

15. Check if an ordered pair is a solution of an equation by putting the x and y values into the equation to see if they make it a true statement.

16. Write the equation. ? 21= 7(3)‏ Substitute 3 for x and 21 for y. ? 21= 21 Checking Solutions of Equations with Two Variables Determine whether the ordered pair is a solution to the given equation. (3, 21); y = 7x y = 7x  So (3, 21) is a solution to y = 7x.

17. Write the equation. ? 20= 5(4)‏ Substitute 4 for x and 20 for y. ? 20= 20 Determine whether the ordered pair is a solution to the given equation. (4, 20); y = 5x y = 5x  So (4, 20) is a solution to y = 5x.

18. You can also graph the solutions of an equation on a coordinate plane. When you graph the ordered pairs of some functions, they form a straight line. The equations that express these functions are called linear equations.

19. Start at the origin and move 4 units right. y Move up until you reach the graph. Move left to find the y-value on the y-axis. x Reading Solutions on Graphs Use the graph of the linear function to find the value of y for the given value of x. x = 4 4 2 -4 -2 0 2 4 -2 When x = 4, y = 2. The ordered pair is (4, 2). -4

20. Start at the origin and move 2 units right. y Move up until you reach the graph. Move left to find the y-value on the y-axis. x Use the graph of the linear function to find the value of y for the given value of x. x = 2 4 2 -4 -2 0 2 4 -2 When x = 2, y = 4. The ordered pair is (2, 4). -4

21. Make a function table. Write these solutions as ordered pairs. x 3x + 2 y 0 3(0) + 2 2 1 3(1) + 2 5 2 3(2) + 2 8 Additional Example 4: Graphing Linear Functions Graph the function described by the equation. y = 3x + 2 (x, y)‏ (0, 2)‏ (1, 5)‏ (2, 8)‏

22. Graph the ordered pairs on a coordinate plane. y 9 Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y. 8 7 6 5 4 3 1 0 x 1 2 3 4 5 6 7 8 9 Additional Example 4 Continued 2

23. Make a function table. Write these solutions as ordered pairs. x 2x + 4 y 0 2(0) + 4 4 1 2(1) + 4 6 2 2(2) + 4 8 Check It Out: Example 4 Graph the function described by the equation. y = 2x + 4 (x, y)‏ (0, 4)‏ (1, 6)‏ (2, 8)‏

24. Graph the ordered pairs on a coordinate plane. y 9 Draw a line through the points to represent all the values of x you could have chosen and the corresponding values of y. 8 7 6 5 4 3 1 0 1 2 3 4 5 6 7 8 9 Check it Out: Example 4 Continued 2 x

25. Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

26. x 0 1 3 5 7 y 0 3 9 15 Lesson Quiz 1. Write an equation for a function that gives the values in the table below. Use the equation to find the value for y for the indicated value of x. 2. Write an equation for the function. Tell what each variable you use represents. The height of a round can is 2 times its radius. y = 3x; 21 h = 2r, where h is the height and r is the radius

27. Lesson Quiz for Student Response Systems 1. Identify an equation for a function that gives the values in the table below. Then, use the equation to find the value for y for the indicated value of x. A. y = 4x + 8; 21 B. y = 7x – 7; 21 C. y = 4x + 8; 28 D. y = 7x – 7; 28

28. Lesson Quiz for Student Response Systems 2. Identify an equation for the function. Tell what each variable you use represents. The width of a swimming pool is twice its depth. A. w = 2d, where d is the width and w is the depth B. , where w is the width and d is the depth C. w = 2d, where w is the width and d is the depth D. , where d is the width and w is the depth

29. Lesson Quiz 1. Use the given x-values to write solutions as ordered pairs to the equation y = 10x + 5 for x = 0, 1, 2, and 3. 2. Determine whether (4, 2) is a solution to the equation y = 5x – 6. (0, 5), (1, 15), (2, 25), (3, 35‏) No, 2 ≠ 5(4) – 6

30. y 9 8 7 6 5 4 3 1 0 x 1 2 3 4 5 6 7 8 9 Lesson Quiz Cont. 3. Graph the function described by the equation y = 5 – x. 2

31. 2. Identify the graph of the function described by the equation y = x – 5. A. B.