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Master Relations and Functions

Learn to analyze and graph relations, find functional values, identify linear equations and functions, and write equations in standard form. Understand the domain, range, and mapping in relations and functions. Practice examples and graphing to test if relations are functions.

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Master Relations and Functions

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  1. Splash Screen

  2. What you will learn today: • Analyze and graph relations • Find functional values • Identify linear equations and functions • Write linear equations in standard form and graph them

  3. 2.1: Relations and Functions • A relation is a set of ordered pairs • The domain of a relation is the set of all first coordinates (x – coordinates) • The range of a relation is the set of all second coordinates (y – coordinates)

  4. 2.1: Relations and Functions • A function is a special type of relation in which each element of the domain is paired with exactly one element of the range; meaning the x – values do not repeat • A mapping shows how each member of the domain with each member of the range.

  5. 2.1: Relations and Functions • A function is called a one – to – one function when each element of the range is paired with exactly one element of the domain; meaning the y – values do not repeat

  6. Example 1-1a State the domain and range of the relation shown in the graph. Is the relation a function? The relation is {(1, 2), (3, 3), (0, –2), (–4, 0), (–3, 1)}. Answer:The domain is {–4, –3, 0, 1, 3}. The range is {–2, 0, 1, 2, 3}. Since no x – value repeats, this is a function.

  7. Example 1-1b State the domain and range of the relation shown in the graph. Is the relation a function? Answer:The domain is {–3, 0, 2, 3}. The range is {–2, –1, 0, 1}. Yes, the relation is afunction.

  8. If no vertical line intersects a graph in more than one point, the graph represents a function Example: If some vertical line intersects a graph in two or more points, the graph does not represent a function Example: Vertical Line Test You can use the vertical line test to determine whether a relation is a function.

  9. Example 1-2a Transportation The table shows the average fuel efficiency in miles per gallon for light trucks for several years. Graph this information and determine whether it represents a function.

  10. Example 1-2b Use the vertical line test. Notice that no vertical line can be drawn that contains more than one of the data points.

  11. Example 1-2d Health The table shows the average weight of a baby for several months during the first year. Graph this information and determine whether it represents a function.

  12. Answer: Yes, this relation is a function. Example 1-2e

  13. 2.1: Relations and Functions • Relations and functions can also be represented by equations • The solution of an equation in x and y are the set of ordered pairs that make the equation true

  14. Graph the relation represented by (2, 5) x y –1 (1, 2) 0 1 (0, –1) 2 (–1, –4) Example 1-3a Make a table of values to find ordered pairs that satisfy the equation. Then graph the ordered pairs. –4 –1 2 5

  15. (2, 5) (1, 2) (0, –1) (–1, –4) Example 1-3b Find the domain and range. Answer: The domain and range are both all real numbers.

  16. (2, 5) Answer:Yes, the equation represents a function. (1, 2) (0, –1) (–1, –4) Example 1-3c Determine whether the relation is a function. Thisgraph passes the vertical line test.

  17. Answer: a. Graphb. Find the domain and range. c. Determine whether the relation is a function. Answer: Yes, the equation is a function. Example 1-3d Answer: The domain and range are both all real numbers.

  18. Graph the relation represented by x y (5, 2) –2 (2, 1) (1, 0) –1 0 (2, –1) 1 (5, –2) 2 Example 1-4a Make a table. In this case, it is easier to choose y values and then find the corresponding values for x. Then sketch the graph, connecting the points with a smoothcurve. 5 2 1 2 5

  19. (5, 2) (2, 1) (1, 0) Answer: The domain is . The range is all real numbers. (2, –1) (5, –2) Example 1-4b Find the domain and range.

  20. x y (5, 2) –2 (2, 1) 5 (1, 0) –1 2 0 1 (2, –1) 1 (5, –2) 2 2 5 Example 1-4c Determine whether the relation is a function. You can see from the table and the vertical line test that there are two y values for each xvalue except x = 1.

  21. Answer: a. Graphb. Find the domain and range. c. Determine whether the relation is a function. Answer: No, the equation does not represent a function. Example 1-4e Answer: The domain is {x|x  –3}. The range is all real numbers.

  22. Independent/Dependent • The domain is also known as the independent variable. It is typically the variable you have no control over • The range is also known as the dependent variable. It is typically the variable that depends on the independent variable; meaning it changes as the independent variable changes

  23. Function Notation • Equations can also be written in function notation • f(x) replaces y in the equation • read “f of x” • f is just the name of the function • to solve a function notation equation, replace what is in () wherever there is an x in the equation, then simplify

  24. Given , find Original function Substitute. Simplify. Answer: Example 1-5a

  25. Given find Original function Substitute. Multiply. Simplify. Answer: Example 1-5b

  26. Given , find Original function Substitute. Answer: Example 1-5c

  27. Given andfind each value. a. b. c. Answer: Example 1-5d Answer: 6 Answer: 0.625

  28. 2.2: Linear Equations • A linear equation has no operations other than addition, subtraction, and multiplication of a variable by a constant. • No multiplication or division of variables • No powers greater than one • The graph is always a line

  29. 2.2: Linear Equations • A linear function is a function whose ordered pairs satisfy a linear equation • Any linear function can be written in the form f(x) = mx + b, where m and b are real numbers

  30. State whether is a linear function. Explain. Answer: This is a linear function because it is in the form Example 2-1a

  31. State whether is a linear function. Explain. Example 2-1b Answer: This is not a linear function because x has an exponent other than 1.

  32. State whether is a linear function. Explain. Answer: This is a linear function because it can be written as Example 2-1c

  33. State whether each function is a linear function. Explain. a. b. c. Answer: yes; Example 2-1d Answer: No; x has an exponent other than 1. Answer: No; two variables are multiplied together.

  34. MeteorologyThe linear function can be usedto find the distance d(s) in miles from a storm, based on the number of seconds s that it takes to hear thunder after seeinglightning. a. If you hear thunder 10 seconds after seeing lightning, howfar away is the storm? b. If the storm is 3 miles away, how long will it take to hear thunderafter seeing lightning? Example 2-2c Answer: 2 miles Answer: 15 seconds

  35. Standard Form • Ax + By = C • A, B, and C are integers • A is positive • No fractions or decimals • x and y are on the same side

  36. Write in standard form. Identify A, B, and C. Example 2-3a

  37. Write in standard form. Identify A, B, and C. Example 2-3b

  38. Write in standard form. Identify A, B, and C. Example 2-3c

  39. Write each equation in standard form. Identify A, B, and C. a. b. c. Answer: and Answer: and and Answer: Example 2-3d

  40. Intercepts • X – intercept: the point where a line crosses the x-axis • Y – intercept: the point where a line crosses the y – axis • Ways to find these: • Cover up method: • If you are trying to find the x – intercept, cover up y and solve what you see; repeat for trying to find the y - intercept • Plug in 0: • If you are trying to find the x – intercept, plug 0 in for y and solve the equation

  41. Find the x-intercept and the y-intercept of the graph of Then graph theequation. Example 2-4a

  42. (0, 4) (–2, 0) Example 2-4c Use the ordered pairs to graph this equation. Answer: The x-intercept is –2, and the y-intercept is 4.

  43. Find the x-intercept and the y-intercept of the graph of Then graph theequation. Answer: The x-intercept is –2, and the y-intercept is 6. Example 2-4d

  44. 2.1/2.2 Practice

  45. Warm - Up • Find the domain and range of x + y = 1. Then determine whether it is a function. • Find f(4) if • Write in standard form. Identify A, B, and C • Find the x – intercept and y – intercept of the graph 3x + 4y = 12. D & R: all real #; yes 5 5x – y = -10; A = 5, B = -1, C = -10 x-intercept is 4; y-intercept is 3

  46. Today you will learn: • Find and use the slope of a line • Graph parallel and perpendicular lines • Write an equation of a line given the slope and a point on the line • Write an equation of a line parallel or perpendicular to a given line

  47. 2.3: Slope • Slope:

  48. Slope formula and Simplify. Example 3-1a Find the slope of the line that passes through (1, 3) and (–2, –3). Then graph the line.

  49. Answer: The slope of the line is Example 3-1c Find the slope of the line that passes through (2, 3) and (–1, 5). Then graph the line.

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