2- 4-’13 What have we reviewed so far? Real Numbers and Their Porperties. Equations and Inequalities with one variable. Functions and Special Functions.

2- 4-’13

• What have we reviewed so far?Real Numbers and Their Porperties.Equations and Inequalities with one

variable.Functions and Special Functions.Inverse Functions.

Review:

2

110 6 9

25

12 3(2)6

2. If w = 4, x = -12, y = 64, z = -3

Find: 5

1

x z y

x

3. Simplify : 12b – 9 + 4b – 7 b +1 =

Review:

4. Solve and graph

a) 4x 1 x 10

b) 13 – (2c + 2) > 2(c + 2) + 3c

5. Solve and check:

a) 3(x – 6) = 4(x + 2) – 21

b) 2(4x + 1) - 2x = 9x - 1

Review 2-5-’13

• Let see the greatest integer function:

6. The Speedy-Fast Parcel Service charges for delivering packages by the weight of the package. If the package weighs less than 1 pound, the cost of delivery is $2. If the package weighs at least 1 pound, but less than 2 pounds, the cost is $3.50. For each additional pound, the cost of delivery increases $1.50. Graph the function that describes this relationship.

• This is an example of an application of the ___________ function. The equation that describes this function is:

f(x) = 1.50[x] + 2

Graph the functionx [x] f(x)

0.1 0 2.00

0.5 0 2.00

0.7 0 2.00

1.0 1 3.50

1.4 1 3.50

1.9 1 3.50

2.4 2 5.00

2.7 2 5.00

3.1 3 6.50

3.7 3 6.50

7. Absolute Value Functions

Graph y = |x| - 3 by completing the t-table:

x y-2 -1 0 1 2

2 4 6–2–4–6 x

2

4

6

–2

–4

–6

y

8. Piecewise Functions

2 is where the graph changes.

Less then 2 uses 3x + 2

Greater then 2 uses x - 3

23

223)(

xx

xxxf

We can and should put in a few x into the function

If f(0) we use 3x + 2, then 3(0) + 2 = 2

If f(3) we use x – 3,

then (3) – 3 = 0

The input tell us what function to use.

23

223)(

xx

xxxf

Piecewise Functions

So put in an x where the domain changes and one point higher

and lower (2, 8)

(2, -1)

Inverse Functions

• Please let’s check the organizers you received on Thursday

Lesson 2.5: Scatter PlotsLesson 2.5: Scatter Plots

Standards: F.IF.7Objective: Determine the correlation of

a scatter plot

F. IF. 5

• Relate the domain of a function to the graph and, whether applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in factory, then the positive integers would be an appropriate domain for the function. Emphasize the selection of a model function based on based on behavior of data and context.

Standards for Mathematical Practices.

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the

reasoning of others.   4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated

reasoning.

Essential Question:

How do I calibrate my axes on a coordinate plane within the context of

the data?

Scatter PlotScatter Plot

• A scatter plot is a graph of a collection of ordered pairs (x,y).

• The graph looks like a bunch of dots, but some of the graphs are a general shape or move in a general direction.

Positive CorrelationPositive Correlation

• If the x-coordinates and the y-coordinates both increase, then it is POSITIVE CORRELATION.

• This means that both are going up, and they are related.

Positive CorrelationPositive Correlation

• If you look at the age of a child and the child’s height, you will find that as the child gets older, the child gets taller. Because both are going up, it is positive correlation.

Age 1 2 3 4 5 6 7 8

Height “

25 31 34 36 40 41 47 55

Negative CorrelationNegative Correlation

• If the x-coordinates and the y-coordinates have one increasing and one decreasing, then it is NEGATIVE CORRELATION.

• This means that 1 is going up and 1 is going down, making a downhill graph. This means the two are related as opposites.

Negative CorrelationNegative Correlation

• If you look at the age of your family’s car and its value, you will find as the car gets older, the car is worth less. This is negative correlation.

Age of car

1 2 3 4 5

Value $30,000 $27,000 $23,500 $18,700 $15,350

No CorrelationNo Correlation

• If there seems to be no pattern, and the points looked scattered, then it is no correlation.

• This means the two are not related.

No CorrelationNo Correlation

• If you look at the size shoe a baseball player wears, and their batting average, you will find that the shoe size does not make the player better or worse, then are not related.

ScatterplotsWhich scatterplots below show a linear trend?

a) c) e)

b) d) f)

NegativeCorrelation

PositiveCorrelation

ConstantCorrelation

Year

Sport Utility Vehicles(SUVs) Sales in U.S.

Sales (in Millions)

19911992

199319941995

1996

19971998

1999

0.91.1

1.41.61.7

2.1

2.42.7

3.2

1991 1993 1995 1997 1999 1992 1994 1996 1998 2000

x

y

Year

Veh

icle

Sal

es (

Mil

lion

s)

5

4

3

2

1

Objective - To plot data points in the coordinate plane and interpret scatter

plots.

1991 1993 1995 1997 1999 1992 1994 1996 1998 2000

x

y

Year

Veh

icle

Sal

es (

Mil

lion

s)

5

4

3

2

1

Trend is increasing.

Scatterplot - a coordinate graph of data points.

Trend appears linear.

Positive correlation.

Year SUV Sales

Predict the sales in 2014.

Plot the data on the graph such that homework timeis on the y-axis and TV time is on the x-axis..

StudentTime SpentWatching TV

Time Spenton Homework

Sam

Jon

Lara

Darren

Megan

Pia

Crystal

30 min.

45 min.

120 min.

240 min.

90 min.

150 min.

180 min.

180 min.

150 min.

90 min.

30 min.

90 min.

90 min.

90 min.

Plot the data on the graph such that homework timeis on the y-axis and TV time is on the x-axis.

TV Homework

30 min.

45 min.

120 min.

240 min.

90 min.

150 min.

180 min.

180 min.

150 min.

90 min.

30 min.

120 min.

120 min.

90 min.

Time Watching TV

Tim

e on

Hom

ewor

k

30 90 150 210 60 120 180 240

240

210

180

150

120

90

60

30

Describe the relationship between time spent onhomework and time spent watching TV.

Time Watching TV

Tim

e on

Hom

ewor

k

30 90 150 210 60 120 180 240

240

210

180

150

120

90

60

30

Trend is decreasing.

Trend appears linear.

Negative correlation.

Time on TV Time on HW

Correlation Coefficient

• The quantity r, called the linear correlation coefficient, measures the strength and the direction of a linear relationship between two variables. The linear correlationcoefficient is sometimes referred to as the Pearson product moment correlation coefficient in honor of its developer Karl Pearson.

Correlation Coefficient

• The value of r is such that -1 < r < +1.  The + and – signs are used for positive linear correlations and negative linear correlations, respectively. 

• A correlation greater than 0.8 is generally described as strong, whereas a correlation less than 0.5 is generally described as weak

Coefficient of Determination r^2

•  The coefficient of determination, r 2, is useful because it gives the proportion of the variance (fluctuation) of one variable that is predictable from the other variable.      It is a measure that allows us to determine how certain one can be in making predictions from a certain model/graph.   The coefficient of determination is the ratio of the explained variation to the total variation. 

Coefficient of Determination r^2

•   The coefficient of determination is such that 0 <  r 2 < 1,  and denotes the strength of the linear association between x and y. 

   The coefficient of determination represents the percent of the data that is the closest to the line of best fit.  For example, if r = 0.922, then r 2 = 0.850, which means that 85% of the total variation in y can be explained by the linear relationship between x and y (as described by the regression equation).  The other 15% of the total variation in y remains unexplained.   The coefficient of determination is a measure of how well the regression line represents the data. 

Interpolation and Extrapolation

Interpolation Interpolation is the process of obtaining a value from a graph or table that is located between major points given, or between data points plotted. A ratio process is usually used to obtain the value.

Extrapolation Extrapolation is the process of obtaining a value from a chart or graph that extends beyond the given data. The "trend" of the data is extended past the last point given and an estimate made of the value.