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~ Chapter 5 ~

Algebra I. Algebra I. Lesson 5-1 Relating Graphs to Events Lesson 5-2 Relations & Functions Lesson 5-3 Function Rules, Tables & Graphs Lesson 5-4 Writing a Function Rule Lesson 5-5 Direct Variation Lesson 5-6 Describing Number Patterns Chapter Review. ~ Chapter 5 ~.

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~ Chapter 5 ~

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  1. Algebra I Algebra I Lesson 5-1 Relating Graphs to Events Lesson 5-2 Relations & Functions Lesson 5-3 Function Rules, Tables & Graphs Lesson 5-4 Writing a Function Rule Lesson 5-5 Direct Variation Lesson 5-6 Describing Number Patterns Chapter Review ~ Chapter 5 ~ Solving & Applying Proportions

  2. Algebra I Algebra I ~ Chapter 5 ~ Cumulative Review Ch 1-4

  3. Lesson 5-1 Interpreting Graphs You can use a graph to show the relationship between two variables… What does each section of the graph represent? Sketching a Graph A plane is flying from NY to London. Sketch a graph of the planes altitude during the flight. Label each section… Relating Graphs to Events Notes

  4. Lesson 5-1 Sketch a graph of the distance from a child’s feet to the ground as the child jumps rope. Label each section. Analyzing Graphs A car travels at a steady speed. Which graph could you use? Relating Graphs to Events Notes B A C

  5. Lesson 5-1 Homework ~ Practice 5-1 Relating Graphs to Events Homework

  6. Lesson 5-2 Relations & Functions Practice 5-1

  7. Lesson 5-2 Relations & Functions Practice 5-1

  8. Lesson 5-2 Relation – a set of ordered pairs The domain of a relation is the first set of coordinates (x values) The range of a relation is the second set of coordinates (y values) Find the domain & range of the relation represented by the data in the table. {-2,-1,4}{-2,1,3} list in order from least to greatest. A function [f(x)] is a relation that assigns exactly one value in the range to each value in the domain. One way to tell whether a relation is a function is to use the vertical-line test. If a vertical line passes through more than one point… the relation is NOT a function. Using a mapping diagram If the domain maps to more than one range… then the relation is not a function. If the domain only maps to one range then the relation is a function. Relations & Functions Notes

  9. Lesson 5-2 A function rule is an equation that describes a function.(Input – x, output – y) Evaluating a function rule… For x = 2.1 y = 2x + 1 f(x) = x2 – 4 g(x) = -x + 2 y = 5.2 f(2.1) = 0.41 g(2.1) = -0.1 Finding the Range You can use a function rule and a given domain to find the range of the function… Find the range of each function for the domain {-2,0,5} f(x) = x – 6 Range = {-8,-6,-1} y = -4x Range = {-20,0,8} g(t) = t2 + 1 y = ¼ x Relations & Functions Notes

  10. Lesson 5-2 Homework ~ Practice 5-2 Relations & Functions Homework

  11. Lesson 5-3 Independent variable – x – the inputs are values for this variable. Dependent variable – y – the outputs are values for this variable. The independent variable graphs on the x-axis, the dependent variable graphs on the y-axis. Model the function rule y = ½ x + 3 using a table of values and a graph… To make a table, choose input values for x and evaluate to find y. To graph, plot points for the ordered pairs from your table… (x,y) Join the points to form a line or a curve. Your turn… Model the function rule f(x) = 3x + 4 A recording company charges $300 for making a master CD and designing the art. It charges $2.50 for burning each CD. Use the function rule P(c) = 300 + 2.5 c. Make a table of values and a graph. Graphing functions Graph the function y = |x| + 1 (hint: make a table of values and then graph) Function Rules, Tables & Graphs Notes

  12. Lesson 5-3 Graph the function f(x) = x2 - 1 Function Rules, Tables & Graphs Notes Homework ~ Practice 5-3

  13. Lesson 5-4 Writing a Function Rule Practice 5-2

  14. Lesson 5-4 Writing a Function Rule Practice 5-3

  15. Lesson 5-4 Writing a Function Rule Practice 5-3

  16. Practice 5-3

  17. Practice 5-3

  18. Lesson 5-4 Writing a rule from a table… Ask what can be done to 1 to get to -1… Then see if that rule applies to get from 2 to 0. If not then try again… So f(x) = x – 2 Rule? y = 2x Write a function rule to calculate the cost of buying apples at $1.25 a pound. Write a function rule to calculate the total distance d(n) traveled after n hours at a constant speed of 45 miles per hour. Write a function rule to calculate the area A(r) of a circle with radius r. Writing a Function Rule Notes

  19. Lesson 5-4 Homework – Practice 5-4 Writing a Function Rule Homework

  20. Lesson 5-5 Direct Variation Practice 5-4

  21. Lesson 5-5 Direct variation – a function in the form ~ y = kx, where k ≠ 0. x & y vary directly… meaning that if x increases in value, y increases in value, and vice versa. The constant of variation, k, is the coefficient of x. Determine if an equation is a direct variation… 5x + 2y = 0 Solve the equation for y 2y = -5x y = -5/2 x (this is in the form y = kx so 5x + 2y = 0 is a direct variation) What is k in 5x + 2y = 0? k = -5/2 ? 7y = 2x 3y + 4x = 8 y – 7.5x = 0 Writing an equation given a point Write an equation of the direct variation that includes the point (4, -3) Remember… y = kx substitute ~ so -3 = k(4) and solve for k k = -3/4 so the equation is y = -3/4 x Direct Variation Notes

  22. Lesson 5-5 Write an equation of the direct variation that includes the point (-3, -6) y = kx… -6 = k(-3) k = 6/3 = 2 so… y = 2x A recipe for one dozen muffins calls for 1 cup of flour. The number of muffins varies directly with the amount of flour you use. Write a direct variation for the relationship between the number of cups of flour and the number of muffins. x = 1 y = 12 so… 12 = k(1) so k = 12 y = 12x Direct Variations & Tables You can rewrite a direct variation y = kx as y/x = k. For each table, use the ratio y/x, to determine whether y varies directly with x. In a direct variation, the ratio is the same for all pairs of data where x ≠ 0. So the proportion x1/y1 = x2/y2 for (x1 ,y1) & (x2, y2) Direct Variation Notes

  23. Lesson 5-5 Homework – Practice 5-5 odd Direct Variation Homework

  24. Lesson 5-6 Describing Number Patterns Practice 5-5

  25. Lesson 5-6 Extending number patterns… What are the next two numbers in the pattern? 1, 4, 9, 16, … rule? 3, 9, 27, 81, … rule? 2, -4, 8, -16, … rule? Each number in a sequence is a term. The common difference (d) is a fixed number in a sequence that is added to each previous term resulting in the next term in the sequence. Determine the common difference in each sequence… 11, 23, 35, 47, … 8, 3, -2, -7, … In a sequence the term is considered to be the output (y). The input(x) is the number of the term in the sequence. Arithmetic sequence = A(n) = a + (n-1)d, where n = term number, a = first term, and d = common difference. Describing Number Patterns Notes

  26. Lesson 5-6 Write the arithmetic sequence for 18, 7, -4, -15,… a = d = so A(n) = + (n - 1) Use the equation to find the 10th term in the sequence above. A(10) = 18 + (10-1)(-11) A(10) = 18 + (-99) A(10) = -81 Your turn.. 1/2 , 1/3, 1/6, 0, … find the 12th term in the sequence. d = - 1/6 so A(n) = 1/2 + (n – 1) (-1/6) and A(12) = 1/2+(12 – 1) (-1/6) A(12) = 1/2+(-11/6) = 3/6 + (-11/6) = -8/6 = -4/3 = -1 1/3 Given the Arithmetic Sequence, find the 3rd, 5th, & 8th term… A(n) = -2.1 + (n – 1)(-5) (plug and chug…) A(3) = -2.1 + (3 – 1)(-5) A(5) = -2.1 + (5 – 1)(-5) A(8) = -2.1 + (8 – 1)(-5) 18 -11 (-11) 18 Describing Number Patterns Notes A(3) = -2.1 + (-10) = -12.1 A(5) = -2.1 + (-20) = -22.1 A(8) = -2.1 + (-35) = -37.1

  27. Lesson 5-6 Homework – Practice 5-6 even Chapter 5 Review due tomorrow Describing Number Patterns Homework

  28. Lesson 5-6 Describing Number Patterns Practice 5-6

  29. Algebra I Algebra I ~ Chapter 5 ~ Chapter Review

  30. Algebra I Algebra I ~ Chapter 5 ~ Chapter Review

  31. Algebra I Algebra I ~ Chapter 5 ~ Chapter Review

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