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Slide 2 2.5 Using Linear Models MonthTemp 1 2 3 4 69 F 70 F 75 F 78 F 1 Slide 3 2.5 Using Linear Models 2 Scatter Plot A graph that relates two sets of data by plotting the data as ordered pairs Slide 4 2.5 Using Linear Models 3 A scatter plot can be used to determine the strength of the relation or the correlation between data sets. The closer the data points fall along a line with a positive slope, The stronger the linear relationship, and the stronger the positive correlation Slide 5 2.5 Using Linear Models 4 STRONG POSITIVE CORRELATION WEAK POSITIVE CORRELATION Describe the correlation shown in each graph. Slide 6 2.5 Using Linear Models 5 STRONG NEGATIVE CORRELATION NO CORRELATION Slide 7 2.5 Using Linear Models Is there a positive, negative, or no correlation between the 2 quantities? If there is a positive or negative correlation, is it strong or weak? 6 Slide 8 2.5 Using Linear Models 7 Age (in years) Height (in feet) A persons age and his height POSITIVE STRONG Slide 9 2.5 Using Linear Models 8 A persons age and the number of cartoons he watches NEGATIVE WEAK Slide 10 2.5 Using Linear Models The table shows the median home prices in New Jersey. An equation is given that models the relationship between time and home prices. Use the equation to predict the median home price in 2010. 9 Slide 11 2.5 Using Linear Models YEAR MEDIAN PRICE ($) 194047,100 195063,100 196076,900 197089,900 1980119,200 1990207,400 2000170,800 10 where x is the number of years since 1940 and y is the price y = 2061x + 47,100 Slide 12 2.5 Using Linear Models 11 y = 2061x + 47,100 y = 2061(70) + 47,100 = $191,370 2010 is 70 years after 1940, so x = 70. The median home price in New Jersey will be approximately $191,370. Slide 13 2.5 Using Linear Models Assignment: p.96-97(#8,12bc,14bc,15-17) For #12 & 14, use these equations. 12.) y = 2053.17x 4,066,574.67 x = year (NOT # of years since 2000) 14.) y = 0.0714x 9.2682 12 Slide 14 2.6 Families of Functions A parent function is the basic starting graph. A transformation is a change to the parent graph. Transformations can be translations or shifts of the graph up or down or left or right. 13 Slide 15 2.6 Families of Functions 14 Examples of transformations Slide 16 2.6 Families of Functions 15 TRANSLATION UP or DOWN Begin with y = f(x). To shift that graph up or down c units, we will write it y = f(x) + c. y = f(x) + 3 y = f(x) 5 Slide 17 2.6 Families of Functions 16 TRANSLATION LEFT OR RIGHT Begin with y = f(x). To shift that graph left or right c units, we will write it y = f(x + c) or y = f(x c). y = f(x 6) y = f(x + 4) Slide 18 2.6 Families of Functions Given the graph of y = f(x), graph y = f(x). 17 + 4 + 4 Slide 19 2.6 Families of Functions Given the graph of y = f(x), graph y = f(x). 18 3 3 Slide 20 2.6 Families of Functions Given the graph of y = f(x), graph y = f(x ). 19 + 4 + 4 Slide 21 2.6 Families of Functions Given the graph of y = f(x), graph y = f(x ). 20 3 3 Slide 22 2.6 Families of Functions Now, if y = f(x), graph y = f(x ). 21 2 2 + 1 + 1 Slide 23 2.6 Families of Functions Assignment: Worksheet (2.6) Translations 22 Slide 24 2.6 Families of Functions 23 ANSWERS TO WORKSHEET 1. f(x + 5) Slide 25 2.6 Families of Functions 24 ANSWERS TO WORKSHEET 2. f(x) 3 Slide 26 2.6 Families of Functions 25 ANSWERS TO WORKSHEET 3. f(x) + 3 Slide 27 2.6 Families of Functions 26 ANSWERS TO WORKSHEET 4. f(x 1) + 2 Slide 28 2.6 Families of Functions 27 ANSWERS TO WORKSHEET 5. f(x + 3) 4 Slide 29 2.6 Families of Functions 28 ANSWERS TO WORKSHEET 6. f(x 5) 3 Slide 30 2.6 Families of Functions More Transformations: Reflection f(x) is a flip of f(x) over the y-axis. f(x) is a flip of f(x) over the x-axis. 29 Slide 31 2.6 Families of Functions More Transformations (continued): Stretch af(x) is a vertical stretch by a factor of a; a > 1 Compression af(x) is a vertical compression by a factor of a; 0 < a < 1 30 Slide 32 2.6 Families of Functions 31 Given y = f(x), graph y = f( x). Slide 33 2.6 Families of Functions 32 Given y = f(x), graph y = f(x). Slide 34 2.6 Families of Functions 33 Given y = f(x), graph y = 2f(x). Slide 35 2.6 Families of Functions 34 Given y = f(x), graph y = 3f(x). Slide 36 2.6 Families of Functions 35 Given y = f(x), graph y = f( x). Slide 37 2.6 Families of Functions 36 Given y = f(x), graph y = 2f(x) + 3. Slide 38 2.6 Families of Functions Assignment: Worksheet (2.6 Enrichment) 37 Slide 39 2.6 Families of Functions 38 ANSWERS ENRICHMENT WORKSHEET 4. y = 2f(x) Slide 40 2.6 Families of Functions 39 ANSWERS ENRICHMENT WORKSHEET 5. y = f(x) 1 Slide 41 2.6 Families of Functions 40 ANSWERS ENRICHMENT WORKSHEET 6. y = f(x + 4) Slide 42 2.6 Families of Functions 41 ANSWERS ENRICHMENT WORKSHEET 7. y = 2f(x + 4) 1 Slide 43 2.6 Families of Functions 42 ANSWERS ENRICHMENT WORKSHEET 8. y = f(x 2) Slide 44 2.6 Families of Functions 43 ANSWERS ENRICHMENT WORKSHEET 9. y = 2f(x) + 1 Slide 45 2.6 Families of Functions 44 ANSWERS ENRICHMENT WORKSHEET 10. y = f(x + 3) 4 Slide 46 2.7 Absolute Value Graphs & Graphs 45 xy 22 11 0 1 2 2 2 1 1 0 Graph f(x) = |x|. Slide 47 Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry? 46 2.7 Absolute Value Graphs & Graphs (0,0) +1and 1 x = 0 Up! Slide 48 2.7 Absolute Value Graphs & Graphs VERTEX FORM OF AN ABSOLUTE VALUE GRAPH 47 Slide 49 The absolute value graph shifts UP if you see + k after the absolute value. The absolute value graph shifts DOWN if you see k after the absolute value. 48 2.7 Absolute Value Graphs & Graphs Slide 50 49 Graph f(x) = |x| + 5. 2.7 Absolute Value Graphs & Graphs Shift the graph of f(x) = |x| UP 5 units!!! Slide 51 Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry? 50 2.7 Absolute Value Graphs & Graphs (0,5) +1and 1 x = 0 Up! Slide 52 2.7 Absolute Value Graphs & Graphs The absolute value graph shifts LEFT h units if you see |x + h| in the equation. The absolute value graph shifts RIGHT h units if you see |x h| in the equation. 51 Slide 53 52 Graph f(x) = |x 4|. 2.7 Absolute Value Graphs & Graphs Shift the graph of f(x) = |x| RIGHT 4 units!!! Slide 54 Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry? 53 2.7 Absolute Value Graphs & Graphs (4,0) +1and 1 x = 4 Up! Slide 55 54 Graph f(x) = |x + 2| + 3. 2.7 Absolute Value Graphs & Graphs Shift the graph of f(x) = |x| LEFT 2 units & UP 3 units!!! Slide 56 Use the previous absolute value graph to answer the questions. What is the vertex? What are the slopes of the rays? What way does the graph open? What is the equation of the axis of symmetry? 55 2.7 Absolute Value Graphs & Graphs ( 2, 3) +1and 1 x = 2 Up! Slide 57 Assignment: p.111(#8 16, 53) For #8 16, do not make a table of values. Shift the parent graph. Use a ruler!!! 56 2.7 Absolute Value Graphs & Graphs Slide 58 The absolute value graph REFLECTS over the x-axis if you see a negative in front of the absolute value. 57 2.7 Absolute Value Graphs & Graphs Slide 59 The absolute value graph is STRETCHED BY A FACTOR OF a if a > 1. The absolute value graph is COMPRESSED BY A FACTOR OF a if 0 < a < 1. 58 2.7 Absolute Value Graphs & Graphs Slide 60 Another way to graph absolute value graphs.. This method is especially useful when a is not 1. 59 2.7 Absolute Value Graphs & Graphs Slide 61 Use f(x) = |x + 2| to find the following information. Vertex: Axis of symmetry: Direction of opening: Slopes of rays: List all transformations. Shift left 2. Compress by a factor of . 60 2.7 Absolute Value Graphs & Graphs ( 2,0) + and x = 2 Up Slide 62 61 Graph f(x) = |x + 2|. 1.) Plot the vertex. V( 2, 0) 2.) Rise and run to get both sides of the V that opens up. 2.7 Absolute Value Graphs & Graphs Slide 63 Use f(x) = 2/3 |x + 3| + 4 to find the following information. Vertex: Axis of symmetry: Direction of opening: Slopes of rays: List all transformations. 62 2.7 Absolute Value Graphs & Graphs ( 3, 4) 2/3 x = 3 Down Shift left 3,shift up 4, reflect over the x-axis, and compress by a factor of 2/3. Slide 64 63 Graph f(x) = 2/3 |x + 3| + 4. 1.) Plot the vertex. V( 3, 4) 2.) Determine whether the V opens up or down. This one: DOWN 3.) Rise and run to get both sides of the V that opens down. 2.7 Absolute Value Graphs & Graphs Slide 65 Use f(x) = 3 |x 5| 3 to find the following information. Vertex: Axis of symmetry: Direction of opening: Slopes of rays: List all transformations. 64 2.7 Absolute Value Graphs & Graphs (5, 3) 3 x = 5 Down Shift right 5,shift down 3, reflect over the x-axis, and stretch by a factor of 3. Slide 66 65 Graph f(x) = 3 |x 5| 3. 1.) Plot the vertex. V(5, 3) 2.) Determine whether the V opens up or down. This one: DOWN 3.) Rise and run to get both sides of the V that opens down. 2.7 Absolute Value Graphs & Graphs Slide 67 66 2.7 Absolute Value Graphs & Graphs Write an absolute value equation for the graph. Slide 68 67 2.7 Absolute Value Graphs & Graphs Write an absolute value equation for the graph. Slide 69 Assignment: p.111(#17 30) For #23 28, find all of the information and then graph. 68 2.7 Absolute Value Graphs & Graphs Slide 70 2.6 Families of Functions 69 Given y = f(x), graph y = 2f(x). Slide 71 2.6 Families of Functions 70