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2.5 Using Linear Models MonthTemp º F 70 º F 75 º F 78 º F 1

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2.5 Using Linear Models 2 Scatter Plot – A graph that relates two sets of data by plotting the data as ordered pairs

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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

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2.5 Using Linear Models 4 STRONG POSITIVE CORRELATION WEAK POSITIVE CORRELATION Describe the correlation shown in each graph.

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2.5 Using Linear Models 5 STRONG NEGATIVE CORRELATION NO CORRELATION

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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

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2.5 Using Linear Models 7 Age (in years) Height (in feet) A person’s age and his height POSITIVE STRONG

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2.5 Using Linear Models 8 A person’s age and the number of cartoons he watches NEGATIVE WEAK

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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

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2.5 Using Linear Models YEAR MEDIAN PRICE ($) , , , , , , , where x is the number of years since 1940 and y is the price y = 2061x + 47,100

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2.5 Using Linear Models 11 y = 2061x + 47,100 y = 2061(70) + 47,100 = $191, is 70 years after 1940, so x = 70. The median home price in New Jersey will be approximately $191,370.

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2.5 Using Linear Models Assignment: p.96-97(#8,12bc,14bc,15-17) For #12 & 14, use these equations. 12.) y = x – 4,066, x = year (NOT # of years since 2000) 14.) y = x –

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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

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2.6 Families of Functions 14 Examples of transformations

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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

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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)

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2.6 Families of Functions Given the graph of y = f(x), graph y = f(x)

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2.6 Families of Functions Given the graph of y = f(x), graph y = f(x). 18 – 3 – 3

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2.6 Families of Functions Given the graph of y = f(x), graph y = f(x )

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2.6 Families of Functions Given the graph of y = f(x), graph y = f(x ). 20 – 3 – 3

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2.6 Families of Functions Now, if y = f(x), graph y = f(x ). 21 – 2 –

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2.6 Families of Functions Assignment: Worksheet (2.6) Translations 22

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2.6 Families of Functions 23 ANSWERS TO WORKSHEET 1. f(x + 5)

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2.6 Families of Functions 24 ANSWERS TO WORKSHEET 2. f(x) – 3

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2.6 Families of Functions 25 ANSWERS TO WORKSHEET 3. f(x) + 3

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2.6 Families of Functions 26 ANSWERS TO WORKSHEET 4. f(x ‒ 1) + 2

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2.6 Families of Functions 27 ANSWERS TO WORKSHEET 5. f(x + 3) ‒ 4

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2.6 Families of Functions 28 ANSWERS TO WORKSHEET 6. f(x ‒ 5) ‒ 3

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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

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2.6 Families of Functions More Transformations (continued): Stretch a∙f(x) is a vertical stretch by a factor of a; a > 1 Compression a∙f(x) is a vertical compression by a factor of a; 0 < a < 1 30

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2.6 Families of Functions 31 Given y = f(x), graph y = f(– x).

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2.6 Families of Functions 32 Given y = f(x), graph y = – f(x).

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2.6 Families of Functions 33 Given y = f(x), graph y = 2f(x).

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2.6 Families of Functions 34 Given y = f(x), graph y = – 3f(x).

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2.6 Families of Functions 35 Given y = f(x), graph y = ½ f(– x).

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2.6 Families of Functions 36 Given y = f(x), graph y = – 2f(x) + 3.

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2.6 Families of Functions Assignment: Worksheet (2.6 Enrichment) 37

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2.6 Families of Functions 38 ANSWERS ENRICHMENT WORKSHEET 4. y = 2f(x)

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2.6 Families of Functions 39 ANSWERS ENRICHMENT WORKSHEET 5. y = f(x) – 1

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2.6 Families of Functions 40 ANSWERS ENRICHMENT WORKSHEET 6. y = f(x + 4)

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2.6 Families of Functions 41 ANSWERS ENRICHMENT WORKSHEET 7. y = 2f(x + 4) – 1

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2.6 Families of Functions 42 ANSWERS ENRICHMENT WORKSHEET 8. y = f(x – 2)

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2.6 Families of Functions 43 ANSWERS ENRICHMENT WORKSHEET 9. y = – 2f(x) + 1

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2.6 Families of Functions 44 ANSWERS ENRICHMENT WORKSHEET 10. y = f(x + 3) – 4

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2.7 Absolute Value Graphs & Graphs 45 xy ‒2‒2 ‒1‒ Graph f(x) = |x|.

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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? Absolute Value Graphs & Graphs (0,0) +1and – 1 x = 0 Up!

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2.7 Absolute Value Graphs & Graphs VERTEX FORM OF AN ABSOLUTE VALUE GRAPH 47

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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 Absolute Value Graphs & Graphs

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49 Graph f(x) = |x| Absolute Value Graphs & Graphs Shift the graph of f(x) = |x| UP 5 units!!!

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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? Absolute Value Graphs & Graphs (0,5) +1and – 1 x = 0 Up!

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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

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52 Graph f(x) = |x – 4|. 2.7 Absolute Value Graphs & Graphs Shift the graph of f(x) = |x| RIGHT 4 units!!!

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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? Absolute Value Graphs & Graphs (4,0) +1and – 1 x = 4 Up!

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54 Graph f(x) = |x + 2| Absolute Value Graphs & Graphs Shift the graph of f(x) = |x| LEFT 2 units & UP 3 units!!!

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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? Absolute Value Graphs & Graphs (– 2, 3) +1and – 1 x = – 2 Up!

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Assignment: p.111(#8 – 16, 53) For #8 – 16, do not make a table of values. Shift the parent graph. Use a ruler!!! Absolute Value Graphs & Graphs

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The absolute value graph REFLECTS over the x-axis if you see a negative in front of the absolute value Absolute Value Graphs & Graphs

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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 < Absolute Value Graphs & Graphs

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Another way to graph absolute value graphs….. This method is especially useful when a is not Absolute Value Graphs & Graphs

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Use f(x) = ½ |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 ½ Absolute Value Graphs & Graphs (– 2,0) + ½ and – ½ x = – 2 Up

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61 Graph f(x) = ½ |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

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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 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.

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63 Graph f(x) = – 2/3 |x + 3| ) 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

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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 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.

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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

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Absolute Value Graphs & Graphs Write an absolute value equation for the graph.

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Absolute Value Graphs & Graphs Write an absolute value equation for the graph.

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Assignment: p.111(#17 – 30) For #23 – 28, find all of the information and then graph Absolute Value Graphs & Graphs

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2.6 Families of Functions 69 Given y = f(x), graph y = 2f(x).

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2.6 Families of Functions 70

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