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Splash Screen. 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. 2.1: Relations and Functions. A relation is a set of ordered pairs

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Splash Screen

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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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