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Chapter 2: Linear equations and functions

Chapter 2: Linear equations and functions

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Chapter 2: Linear equations and functions

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  1. Chapter 2:Linear equations and functions BIG IDEAS: Representing relations and functions Graphing linear equations and inequalities in two variables Writing linear equations and inequalities in two variables

  2. On a blank notebook page, please complete the Prerequisite Skills on PG 70 #2-12 Even

  3. Lesson 1: represent relations and functions

  4. Essential question How do you graph relations and functions?

  5. Relation: A mapping, or pairing, of input values with output values • Domain: The set of input values of a relation • Range: The set of output values of a relation • Function: A relation for which each input has exactly one output VOCABULARY

  6. The domain consists of all the x-coordinates:–2, –1, 1, 2,and3. The rangeconsists of all the y-coordinates:–3, –2, 1, and 3. EXAMPLE 1 Represent relations Consider the relation given by the ordered pair (–2, –3), (–1, 1), (1, 3), (2, –2), and (3, 1). a.Identify the domain and range. SOLUTION

  7. b. Represent the relation using a graph and a mapping diagram. EXAMPLE 1 Represent relations SOLUTION b. Graph Mapping Diagram

  8. The relation isa function because each input is mapped onto exactly one output. a. Identify functions EXAMPLE 2 Tell whether the relation is a function. Explain. SOLUTION

  9. b. The relation isnota function because the input 1 is mapped onto both – 1 and 2. Identify functions EXAMPLE 2 Tell whether the relation is a function. Explain. SOLUTION

  10. for Examples 1 and 2 GUIDED PRACTICE 1. Consider the relation given by the ordered pairs (–4, 3), (–2, 1), (0, 3), (1, –2), and (–2, –4) a. Identify the domain and range. SOLUTION The domain consists of all the x-coordinates:–4, –2, 0 and 1, The rangeconsists of all the y-coordinates: 3, 1,–2 and –4

  11. for Examples 1 and 2 GUIDED PRACTICE b. Represent the relation using a table and a mapping diagram. SOLUTION

  12. 2. Tell whether the relation is a function. Explain. ANSWER Yes; each input has exactly one output. for Examples 1 and 2 GUIDED PRACTICE

  13. Use the vertical line test EXAMPLE 3 SOLUTION The team graph does not represent a function because vertical lines at x=28 and x=29 each intersect the graph at more than one point. The graph for Kevin Garnett does represent a function because no vertical line intersects the graph at more than one point.

  14. Graph an equation in two variables EXAMPLE 4 Graph the equationy= – 2x–1. SOLUTION STEP1 Construct a table of values.

  15. Graph an equation in two variables EXAMPLE 4 STEP 2 Plot the points. Notice that they all lie on a line. STEP3 Connect the points with a line.

  16. a. f (x) = –x2 – 2x + 7 The functionfis not linear because it has an x2-term. EXAMPLE 5 Classify and evaluate functions Tell whether the function is linear.Thenevaluate the function when x= – 4. SOLUTION f (x) =–x2– 2x+ 7 Write function. f (–4) =–(– 4)2– 2(–4) + 7 Substitute–4forx. Simplify. =–1

  17. The function gis linear because it has the form g(x) = mx + b. b. g(x) = 5x + 8 EXAMPLE 5 Classify and evaluate functions SOLUTION g(x) = 5x+ 8 Write function. Substitute –4 forx. g(–4) = 5(–4) + 8 Simplify. =–12

  18. for Examples 4 and 5 GUIDED PRACTICE 4. Graph the equationy = 3x – 2. ANSWER

  19. for Examples 4 and 5 GUIDED PRACTICE Tell whether the function is linear. Then evaluate the function when x = –2. 5. f (x) = x – 1 – x3 6. g (x) = –4 – 2x ANSWER ANSWER Not Linear; Thef(x) = 5,whenx = –2 Linear; Thef(x) = 0,whenx = –2

  20. Essential question How do you graph relations and functions? Make a table of domain and range values. Then plot the points from the table. If it is a function, connect the dots.

  21. An internet company had a profit of $2.6 million in retail sales over the last five years. What was its average annual profit?

  22. Lesson 2: Find Slope and Rate of change

  23. Essential question How do you determine whether two nonvertical lines are parallel or perpendicular?

  24. Slope: The ratio of vertical change (the rise) to the horizontal change (the run) for a nonvertical line • Parallel: Two lines in the same plane that do not intersect • Perpendicular: Two lines in the same plane that intersect to form a right angle • Rate of change: A comparison of how much one quantity changes, on average, relative to the change in another quantity. VOCABULARY

  25. A skateboard ramp has a rise of 15 inches and a run of 54 inches. What is its slope? rise slope = run 5 5 15 18 18 = = 54 ANSWER . The slope of the ramp is EXAMPLE 1 Find slope in real life Skateboarding SOLUTION

  26. m = y2–y1 –1 –3 = x2–x1 2–(–1) 4 ANSWER 3 = The correct answer is A. EXAMPLE 2 Standardized Test Practice SOLUTION Let (x1, y1) =(–1, 3) and (x2,y2)=(2, –1).

  27. 2. What is the slope of the line passing through the points (–4, 9) and (–8, 3) ? ANSWER The correct answer is D. for Examples 1 and 2 GUIDED PRACTICE

  28. 1 5 1 1 – 2 2 3 4 ANSWER ANSWER ANSWER ANSWER for Examples 1 and 2 GUIDED PRACTICE Find the slope of the line passing through the given points. 3. (0, 3), (4, 8) 5. (–3, –2), (6, 1) 4. (– 5, 1), (5, – 4) 6. (7, 3), (–1, 7)

  29. a. Line 1: through (–2, 2) and (0, –1) Line 2: through (–4, –1) and (2, 3) b. Line 1: through (1, 2) and (4, –3) Line 2: through (–4, 3) and (–1, –2) –1 – 2 = a. Find the slopes of the two lines. 0– (–2) 3 – 3 2 = – m1 = 2 EXAMPLE 4 Classify parallel and perpendicular lines Tell whether the lines are parallel, perpendicular, or neither. SOLUTION

  30. 11.Line1: through(–2, 8)and(2, –4) 12.Line1: through(–4, –2)and(1, 7) Line2:through(–5, 1)and(–2, 2) Line2:through(–1, –4)and(3, 5) for Example 4 GUIDED PRACTICE GUIDED PRACTICE Tell whether the lines are parallel, perpendicular, or neither. ANSWER perpendicular ANSWER neither

  31. Essential question How do you determine whether two nonvertical lines are parallel or perpendicular? Calculate the slope: Parallel lines have equal slope Perpendicular lines have slopes that are opposite reciprocals

  32. In 2005, Carey’s Pet Shop had a profit of $55,500. In 2006, profits were $38,700. In a graph of the data, is the slope of the segment between 2005 and 2006 positive or negative?

  33. Lesson 3: Graph equations of lines

  34. Essential question How do you graph a linear equation using intercepts?

  35. Y-intercept: The y-coordinate of a point where a graph intersects the y-axis • X-intercept: The x-coordinate of a point where a graph intersects the x-axis • Slope-intercept form: y=mx+b • Standard form: Ax + By = C where a ≠ 0 VOCABULARY

  36. b. a. y = x + 3 y = 2x a. EXAMPLE 1 Graph linear functions Graph the equation. Compare the graph with the graph of y = x. SOLUTION The graphs ofy = 2x andy = x both have a y-intercept of 0, but the graph ofy = 2x has a slope of2 instead of 1.

  37. b. EXAMPLE 1 Graph linear functions The graphs ofy = x + 3 and y = x both have a slope of 1, but the graph ofy = x + 3has a y-intercept of 3 instead of 0.

  38. EXAMPLE 2 Graph an equation in slope-intercept form STEP 4 Draw a line through the two points.

  39. 2 5 5. y = x + 4 for Examples 1 and 2 GUIDED PRACTICE Graph the equation 4. y = –x + 2

  40. 1 2 6. y = x – 3 for Examples 1 and 2 GUIDED PRACTICE Graph the equation 7. y = 5 + x

  41. for Examples 1 and 2 GUIDED PRACTICE Graph the equation 8. f (x) = 1 – 3x 9. f (x) = 10 – x

  42. EXAMPLE 3 Solve a multi-step problem Biology The body length y (in inches) of a walrus calf can be modeled by y = 5x + 42 where xis the calf’s age (in months). • Graph the equation. • Describe what the slope and y-intercept represent in this situation. • Use the graph to estimate the body length of a calf that is 10 months old.

  43. EXAMPLE 3 Solve a multi-step problem SOLUTION STEP 1 Graph the equation. STEP 2 Interpret the slope and y-intercept. The slope, 5,represents the calf’s rate of growth in inches per month. The y-intercept, 42, represents a newborn calf’s body length in inches.

  44. EXAMPLE 4 Graph an equation in standard form STEP3 Identify the y-intercept. Lety= 0. 5(0) + 2y = 10 y = 5 Solve for y. The y-intercept is 5. So, plot the point (0, 5). STEP4 Draw a line through the two points.

  45. EXAMPLE 5 Graph horizontal and vertical lines Graph (a) y = 2 and (b) x = –3. SOLUTION a. The graph of y = 2 is the horizontal line that passes through the point (0, 2). Notice that every point on the line has a y-coordinate of 2. b. The graph of x = –3 is the vertical line that passes through the point (–3, 0). Notice that every point on the line has an x-coordinate of –3.

  46. for Examples 4 and 5 GUIDED PRACTICE Graph the equation. 12. 3x – 2y = 12 11. 2x + 5y = 10

  47. for Examples 4 and 5 GUIDED PRACTICE Graph the equation. 13. x = 1 14. y = –4

  48. Essential question How do you graph a linear equation using intercepts? To find x: Set y = 0, solve for x, (x,0) To find y: Set x = 0, solve for y, (0,y) Plot each intercept and connect the dots.

  49. On a blank piece of paper please complete the Quiz for Lessons 2.1-2.3 on Page 96 #1-9. When finished please turn into the homework bin.

  50. Lesson 4: Write equations of lines