2-5 Using Linear Models

Quick Review

You can use a scatter plot to show relationships between data sets. You can make predictions using a trend line, which approximates the relationship between two data sets. The most accurate trend line is a line of best fit.

Example

Draw a scatter plot of the data. Is a linear model reasonable? If so, predict the value of y when x = 9 . x equals 9 .

{(0, 6), (1, 7), (2, 5), (3, 4), (4, 2), (5, 1)}

The points are close to the line y = − 4 3 x + 8 , y equals negative , 4 thirds , x plus 8 comma  so a linear model is reasonable. When x = 9 , x equals 9 comma

y = − 4 3 ( 9 ) + 8   = − 4 table with 2 rows and 2 columns , row1 column 1 , y , column 2 equals negative , 4 thirds , open 9 close plus 8 , row2 column 1 , , column 2 equals negative 4 , end table

Exercises

Draw a scatter plot of each set of data. Decide whether a linear model is reasonable. If so, describe the correlation. Then draw a trend line and write its equation. Predict the value of y when x is 15.

31. {(3, 5), (4, 7), (5, 9), (7, 10), (8, 10), (9, 11), (10, 13)}
32. {(6, 15.5), (7, 14.0), (8, 13.0), (9, 12.5), (10, 12.0), (11, 11.5), (12, 10.0)}

33. x	0	3	6	9	12
    y	17.5	35.4	50.5	60.6	66.3

2-6 Families of Functions

Quick Review

A parent function is the simplest form of a function in a family of functions. Each member is a transformation of the parent function.

Translations shift the graph horizontally, vertically, or both. A reflection flips the graph over a line of symmetry. Vertical stretches and compressions change the shape of the graph by a factor.

Example

Write the equation of the transformation of the graph of f ( x ) = x 2 f open x close equals , x squared  translated 3 units up, vertically stretched by a factor of 6, and reflected across the y-axis.

y = x 2 + 3 Translated  3  units up. y = 6 ( x 2 + 3 ) Vertically stretched. y = 6 ( − x ) 2 + 18 Reflected across the  y -axis. y = 6 x 2 + 18   table with 4 rows and 3 columns , row1 column 1 , y , column 2 equals , x squared , plus 3 , column 3 cap translated . 3 . unitsup. , row2 column 1 , y , column 2 equals 6 open , x squared , plus 3 close , column 3 cap verticallystretched. , row3 column 1 , y , column 2 equals 6 . open negative x close squared . plus 18 , column 3 cap reflectedacrossthe . y , minusaxis. , row4 column 1 , y , column 2 equals , 6 x squared , plus 18 , column 3 , end table

Exercises

Write the equation for the transformation of the graph of y = f (x).

34. translated 2 units left, 7 units down
35. translated 5 units right, reflected across the x-axis
36. translated 3 units up, reflected across the y-axis

Describe the transformation(s) of the parent function f(x).

37. g ( x ) = f ( x ) − 4 g open x close equals f open x close minus 4
38. h(x) = 12f(x) + 2
39. k ( x ) = − 2 f ( − x ) k open x close equals negative 2 f open negative x close

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