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Understanding Functions and Their Applications in Everyday Contexts

This lesson covers the concept of functions, exploring their definitions, domains, ranges, and the construction of function tables. Students will learn to interpret functions through examples like parking fees and taxi fares, enabling them to create functions using variables and apply them to real-life situations. The lesson also includes exercises on finding function values, graphing functions, and analyzing slopes (positive, negative, zero, and undefined) to deepen their understanding of linear relationships.

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Understanding Functions and Their Applications in Everyday Contexts

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  1. Splash Screen

  2. Ch. 11-2 Functions

  3. Vocabulary: • Function: a relationship where one thing depends upon another. E.g. f(x) = 7x (textbook p.517) • f(x) is read as function of x. E.g. f(4) is read as function of 4. • Function table: a table that organize the input, rule, and output of a function • Domain: the set of input values • Range: the set of output values

  4. For example: f(x) = 4x – 1 is a function. Below is its function table. f(-3) = 4(-3) -1 = -12 – 1 = -13 Domain Range

  5. Find the function value of Example 2-1a Substitute 4 for x into the function rule. Answer:

  6. Find the function value of Example 2-1b Answer: –5

  7. Find the function value of Answer: Example 2-2a Substitute –6 for x into the function rule. Simplify.

  8. Find the function value of Example 2-2b Answer: 2

  9. Complete the function table for Example 2-3a Substitute each value of x, or input, into the function rule. Then simplify to find the output.

  10. Inputx Rule 4x – 1 Output f(x) 0 1 Example 2-3b Answer:

  11. Complete the function table for Example 2-3c Answer:

  12. Answer: The function represents the situation. Example 2-4a PARKING FEESThe price for parking at a city lot is $3.00 plus $2.00 per hour. Write a function using two variables to represent the price of parking for h hours. Words Cost of parking equals $3.00 plus $2.00 per hour. Function

  13. Answer: Example 2-4b TAXIThe price for a taxi ride is $5.00 plus $4.00 per hour. Write a function using two variables to represent the price of riding a taxi for h hours.

  14. Example 2-5a PARKING FEESThe price for parking at a city lot is $3.00 plus $2.00 per hour. How much would it cost to park at the lot for 2 hours? Substitute 2 for h into the function rule. Answer: It will cost $7.00 to park for 2 hours.

  15. Example 2-5b TAXIThe price for a taxi ride is $5.00 plus $1.00 per hour. How much would it cost for a 3 hour taxiride? Answer: $17.00

  16. End of Lesson 2

  17. Lesson 3 Contents Example 1Graph a Function Example 2Use x- and y-intercepts

  18. Graph x x – 3 y (x, y) (0, –3) 0 –3 0 – 3 1 1 – 3 –2 (1, –2) 2 2 – 3 –1 (2, –1) 3 3 – 3 0 (3, 0) Example 3-1a Step 1 Choose some values for x. Make a function table. Include a column of ordered pairs of the form (x, y).

  19. Note that the ordered pair for any point on this line is a solution of The line is the complete graph of the function. Answer: y = x – 3 (3, 0) (2, –1) (1, –2) (0, –3) Example 3-1b Step 2 Graph each ordered pair. Draw a line that passes through each point.

  20. Example 3-1b Check It appears from the graph that (–1, –4) is also a solution. Check this by substitution. Write the function. Simplify.

  21. Graph Example 3-1c Answer:

  22. MULTIPLE-CHOICE TEST ITEMWhich graph represents A B C D Example 3-2a

  23. Read the Test Item You need to decide which of the four graphs represents The graph will cross the x-axis when Example 3-2b Solve the Test Item Replace y with 0. Subtract 1. Simplify. Divide by 2. Simplify.

  24. The graph will cross the y-axis when The x-intercept is and the y-intercept is1. Graph D is the only graph with both of these intercepts. Example 3-2c Replace x with 0. Simplify. Simplify. Answer: D

  25. MULTIPLE-CHOICE TEST ITEMWhich graph represents Example 3-2e Answer: C

  26. End of Lesson 3

  27. Lesson 4 Contents Example 1Positive Slope Example 2Negative Slope Example 3Zero Slope Example 4Undefined Slope

  28. Ch. 11-4 Slope Formula Vocabularies/ concepts: Slope: the ratio of the rise to the run. Slope = rise run m = y2 – y1 x2 – x1 Positive slope: E.g.(3,4) (-1,2)

  29. Ch. 5-1 Slope Negative slope: E.g. (-4,1) (-1,-2) Zero slope: E.g. (-1,2) (1,2) Undefined slope: E.g. (1,3) (1,-2)

  30. Example 4-1a Find the slope of the line that passes through A(3, 3) and B(2, 0). Definition of slope Simplify.

  31. Example 4-1b CheckWhen going from left to right, the graph of the line slants upward. This is consistent with a positive slope. Answer: The slope is 3.

  32. Example 4-1c Find the slope of the line that passes through A(4, 3) and B(1, 0). Answer: 1

  33. Example 4-2a Find the slope of the line that passes through X(–2, 3) and Y(3, 0). Definition of slope Simplify.

  34. Answer: The slope is . Example 4-2b CheckWhen going from left to right, the graph of the line slants downward. This is consistent with a negative slope.

  35. Answer: Example 4-2c Find the slope of the line that passes through X(–3, 3) and Y(1, 0).

  36. Example 4-3a Find the slope of the line that passes through P(6, 5) and Q(2, 5). Definition of slope Simplify.

  37. Example 4-3b Answer: The slope is 0. The slope of any horizontal line is 0.

  38. Example 4-3c Find the slope of the line that passes through P(1, 6) and Q(2, 6). Answer: 0

  39. Example 4-4a Find the slope of the line that passes through G(2, 4) and H(2, 6). Definition of slope Simplify.

  40. Example 4-4b Answer: Division by 0 is not defined. So, the slope is undefined. The slope of any vertical line is undefined.

  41. Example 4-4c Find the slope of the line that passes through G(–2, 1) and H(–2, 0). Answer: undefined

  42. End of Lesson 4

  43. Lesson 5 Contents Example 1Find Slopes and y-intercepts of Graphs Example 2Find Slopes and y-intercepts of Graphs Example 3Graph an Equation Example 4Graph an Equation to Solve Problems Example 5Graph an Equation to Solve Problems Example 6Graph an Equation to Solve Problems

  44. Ch.11-5 Vocabularies/ concepts: Slope-intercept form: y = mx + b So, if y = 2x + 5 What is the slop? What is the y-intercept? Slope y-intercept

  45. Write an equation of the line whose slope is and whose y-intercept is –6. Slope-intercept form Replace m with and b with –6. Answer: Example 3-1a

  46. Ch. 5-3 Slope-intercept Form Vocabularies/ concepts: Slope-intercept form: Y = mx + b So, with one coordinate and the slope, we can write the slope-intercept form and grape it. Slope y-intercept

  47. Answer: Example 3-1b Write an equation of the line whose slope is 4and whose y-intercept is 3.

  48. State the slope and the y-intercept of the graph of the equation of Write the equation in the form Answer: The slope of the graph is and the y-intercept is –5. Example 5-1a

  49. State the slope and the y-intercept of the graph of the equation of Example 5-1b Answer:

  50. State the slope and the y-intercept of the graph of the equation of Write the equation in the form Example 5-2a Write the original equation. Subtract 2x from each side. Simplify. Answer: The slope of the graph is –2 and the y-intercept is 8.

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