2.5 Using Linear Models   Month Temp 1 2 3 4 69 º F 70 º F 75 º F 78 º F

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

2.5 Using Linear Models 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

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

2.5 Using Linear Models STRONG NEGATIVE CORRELATION NO CORRELATION

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?

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

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

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.

2.5 Using Linear Models y = 2061x + 47,100 YEAR MEDIAN PRICE ($) 1940 47,100 1950 63,100 1960 76,900 1970 89,900 1980 119,200 1990 207,400 2000 170,800 y = 2061x + 47,100 where x is the number of years since 1940 and y is the price

2.5 Using Linear Models y = 2061x + 47,100 2010 is 70 years after 1940, so x = 70. y = 2061(70) + 47,100 = $191,370 The median home price in New Jersey will be approximately $191,370.

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

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.

Examples of transformations 2.6 Families of Functions Examples of transformations

2.6 Families of Functions 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

2.6 Families of Functions 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 + 4) y = f(x − 6)

2.6 Families of Functions Given the graph of y = f(x), graph y = f(x) . + 4 + 4

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

2.6 Families of Functions Given the graph of y = f(x), graph y = f(x ). + 4 + 4

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

2.6 Families of Functions Now, if y = f(x), graph y = f(x ) . – 2 – 2 + 1 + 1

2.6 Families of Functions Assignment: Worksheet (2.6) Translations

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

2.6 Families of Functions Assignment: Worksheet (2.6 Enrichment)

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

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

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

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

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

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

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

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

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

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

2.7 Absolute Value Graphs & Graphs 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.

2.7 Absolute Value Graphs & Graphs Graph f(x) = |x| + 5 . Shift the graph of f(x) = |x| UP 5 units!!! Remember to keep the slopes of the rays +1 and – 1!!!

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

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.

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

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

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

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

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

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

2.7 Absolute Value Graphs & Graphs 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.

2.7 Absolute Value Graphs & Graphs Another way to graph absolute value graphs….. This method is especially useful when a is not 1.

2.7 Absolute Value Graphs & Graphs 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. (– 2,0) x = – 2 Up + ½ and – ½

2.7 Absolute Value Graphs & Graphs 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 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. (– 3, 4) x = – 3 Down ± 2/3 Shift left 3, shift up 4, reflect over the x-axis, and compress by a factor of 2/3.

2.7 Absolute Value Graphs & Graphs 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 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. (5, – 3) x = 5 Down ± 3 Shift right 5, shift down 3, reflect over the x-axis, and stretch by a factor of 3.

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

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

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

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

2.6 Families of Functions 70