Algebra 2 Chapter

2.5 Using Linear ModelsMonthTemp123469 F70 F75 F78 F1

2.5 Using Linear Models2Scatter Plot A graph that relates two sets of data by plotting the data asordered pairs2.5 Using Linear Models3A 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

2.5 Using Linear Models4STRONGPOSITIVE CORRELATIONWEAKPOSITIVE CORRELATIONDescribe the correlation shown in each graph.

42.5 Using Linear Models5STRONG NEGATIVE CORRELATIONNO CORRELATION2.5 Using Linear ModelsIs there a positive, negative, or no correlation between the 2 quantities?

If there is a positive or negative correlation, is it strong or weak?62.5 Using Linear Models7Age (in years)Height (in feet)A persons age and his heightPOSITIVESTRONG2.5 Using Linear Models8

A persons age and the number of cartoons he watchesNEGATIVEWEAK2.5 Using Linear ModelsThe 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 2010.92.5 Using Linear ModelsYEARMEDIAN PRICE ($)194047,100195063,100196076,900197089,9001980119,2001990207,4002000170,80010 where x is the number of years since 1940 and y is the pricey = 2061x + 47,100

2.5 Using Linear Models11y = 2061x + 47,100y = 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.

2.5 Using Linear ModelsAssignment:p.96-97(#8,12bc,14bc,15-17)

For #12 & 14, use these equations.12.) y = 2053.17x 4,066,574.67x = year (NOT # of years since 2000) 14.) y = 0.0714x 9.2682

122.6 Families of FunctionsA 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.132.6 Families of Functions14

Examples of transformations2.6 Families of Functions15

TRANSLATION UP or DOWNBegin with y = f(x).

To shift that graph up or down c units, we will write it y = f(x) + c.

y = f(x) + 3y = f(x) 52.6 Families of Functions16

TRANSLATION LEFT OR RIGHTBegin 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)2.6 Families of FunctionsGiven the graph of y = f(x), graph y = f(x) .17

+ 4+ 42.6 Families of FunctionsGiven the graph of y = f(x), graph y = f(x) .18

3 32.6 Families of FunctionsGiven the graph of y = f(x), graph y = f(x ).19

+ 4+ 42.6 Families of FunctionsGiven the graph of y = f(x), graph y = f(x ).20

3 32.6 Families of FunctionsNow, if y = f(x), graph y = f(x ) .21

2 2+ 1+ 12.6 Families of FunctionsAssignment:Worksheet (2.6) Translations222.6 Families of Functions23

ANSWERS TO WORKSHEET1.f(x + 5)2.6 Families of Functions24

ANSWERS TO WORKSHEET2.f(x) 3 2.6 Families of Functions25

ANSWERS TO WORKSHEET3.f(x) + 3 2.6 Families of Functions26

ANSWERS TO WORKSHEET4.f(x 1) + 2 2.6 Families of Functions27

ANSWERS TO WORKSHEET5.f(x + 3) 4 2.6 Families of Functions28

ANSWERS TO WORKSHEET6.f(x 5) 3 2.6 Families of FunctionsMore Transformations:Reflectionf(x) is a flip of f(x) over the y-axis. f(x) is a flip of f(x) over the x-axis.

292.6 Families of FunctionsMore Transformations (continued):Stretchaf(x) is a vertical stretch by a factor of a; a > 1Compressionaf(x) is a vertical compression by a factor of a; 0 < a < 1

302.6 Families of Functions31

Given y = f(x), graph y = f( x).2.6 Families of Functions32

Given y = f(x), graph y = f(x).2.6 Families of Functions33

Given y = f(x), graph y = 2f(x).2.6 Families of Functions34

Given y = f(x), graph y = 3f(x).2.6 Families of Functions35

Given y = f(x), graph y = f( x).2.6 Families of Functions36

Given y = f(x), graph y = 2f(x) + 3.2.6 Families of FunctionsAssignment:Worksheet (2.6 Enrichment)372.6 Families of Functions38ANSWERS ENRICHMENT WORKSHEET4.

y = 2f(x)2.6 Families of Functions39ANSWERS ENRICHMENT WORKSHEET5.

y = f(x) 1 2.6 Families of Functions40ANSWERS ENRICHMENT WORKSHEET6.

y = f(x + 4) 2.6 Families of Functions41ANSWERS ENRICHMENT WORKSHEET7.

y = 2f(x + 4) 12.6 Families of Functions42ANSWERS ENRICHMENT WORKSHEET8.

y = f(x 2) 2.6 Families of Functions43ANSWERS ENRICHMENT WORKSHEET9.

y = 2f(x) + 12.6 Families of Functions44ANSWERS ENRICHMENT WORKSHEET10.

y = f(x + 3) 42.7 Absolute Value Graphs & Graphs45

xy2101222110Graph f(x) = |x|.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? 462.7 Absolute Value Graphs & Graphs(0,0)+1 and 1 x = 0Up!2.7 Absolute Value Graphs & GraphsVERTEX FORM OF AN ABSOLUTE VALUE GRAPH

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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.

482.7 Absolute Value Graphs & Graphs49

Graph f(x) = |x| + 5 .2.7 Absolute Value Graphs & GraphsShift the graph of f(x) = |x|UP 5 units!!!

Remember to keep the slopes of the rays +1 and 1!!!

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? 502.7 Absolute Value Graphs & Graphs(0,5)+1 and 1 x = 0Up!2.7 Absolute Value Graphs & GraphsThe 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.

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Graph f(x) = |x 4| . 2.7 Absolute Value Graphs & GraphsShift the graph of f(x) = |x|Remember to keep the slopes of the rays +1 and 1!!!

RIGHT 4 units!!!

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? 532.7 Absolute Value Graphs & Graphs(4,0)+1 and 1 x = 4Up!54

Graph f(x) = |x + 2| + 3 . 2.7 Absolute Value Graphs & GraphsShift the graph of f(x) = |x|LEFT 2 units & UP 3 units!!!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? 552.7 Absolute Value Graphs & Graphs( 2, 3)+1 and 1 x = 2Up!Assignment:p.111(#8 16, 53)For #8 16, do not make a table of values. Shift the parent graph.Use a ruler!!!

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

572.7 Absolute Value Graphs & GraphsThe 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.

582.7 Absolute Value Graphs & GraphsAnother way to graph absolute value graphs..This method is especially useful when a is not 1.592.7 Absolute Value Graphs & GraphsUse 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 . 602.7 Absolute Value Graphs & Graphs( 2,0)+ and x = 2Up61

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 & GraphsUse f(x) = 2/3 |x + 3| + 4to find the following information.Vertex:Axis of symmetry:Direction of opening: Slopes of rays:List all transformations. 622.7 Absolute Value Graphs & Graphs( 3, 4) 2/3 x = 3DownShift left 3, shift up 4, reflect over the x-axis,and compress by a factor of 2/3.63

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: DOWN3.) Rise and run to get both sides of the V that opens down.2.7 Absolute Value Graphs & GraphsUse f(x) = 3 |x 5| 3to find the following information.Vertex:Axis of symmetry:Direction of opening: Slopes of rays:List all transformations. 642.7 Absolute Value Graphs & Graphs(5, 3) 3 x = 5DownShift right 5, shift down 3, reflect over the x-axis,and stretch by a factor of 3.65

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: DOWN3.) Rise and run to get both sides of the V that opens down.2.7 Absolute Value Graphs & Graphs66

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

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

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

682.7 Absolute Value Graphs & Graphs2.6 Families of Functions69

Given y = f(x), graph y = 2f(x).2.6 Families of Functions70