5-Minute Check on Activity 4-6

5-Minute Check on Activity 4-65-Minute Check on Activity 4-65-Minute Check on Activity 4-65-Minute Check on Activity 4-6

Click the mouse button or press the Space Bar to display the answers.Click the mouse button or press the Space Bar to display the answers.

List the quadratic formula:

1.

Set up the following (do not solve) using the quadratic formula

2. y = 2x2 – 3x + 4

3. y = 3x2 – 4x – 5

4. y = x2 – 3x + 4

Solve:

5. y = 4x2 + 7x – 2

-b ± b² – 4ac x = ------------------------- 2a

-(-3) ± (-3)² – 4(2)(4) x = -------------------------------- 2(2)

-(-4) ± (-4)² – 4(3)(-5) x = -------------------------------- 2(3)

-(-3) ± (-3)² – 4(1)(4) x = -------------------------------- 2(1)

-(7) ± (7)² – 4(4)(-2) -7 ± 81x = -------------------------------- = ------------ = ¼ , -2 2(4) 8

Activity 4 - 7

Air Quality in Atlanta

Objectives• Determine quadratic regression models using the

graphing calculator

• Solve problems using quadratic regression models

Vocabulary• None new

Activity

The Air Quality Index (AQI) measures how polluted the air is by measuring five major pollutants: ground-level ozone, particulate matter, carbon monoxide, sulfur dioxide, and nitrogen oxide. Based on the amount of each pollutant in the air, the AQI assigns a numerical value to air quality, as follows:

Numerical Rating Meaning

0 – 50 Good

51 – 100 Moderate

101 – 150 Unhealthy for sensitive groups

151 – 200 Unhealthy

201 – 300 Very unhealthy

301 – 500 Hazardous

Activity contThe following table indicates the number of days in which the AQI was greater than 100 in the city of Atlanta, Georgia.

Sketch a scatterplot of the data. Let t represent the number of years since 1990 (t in L1 and Days in L2). Therefore, t = 0 corresponds to the year 1990.

Year 1990 1992 1994 1996 1998 1999

Days AQI > 100 42 20 15 25 50 61

D

t

What does the shape of the graph look like?

parabolic or quadratic

Activity contIf the shape was linear, we used LinReg on our calculator to make a model of the data.

Our calculator has several other regression models in the STAT, CALC menu. The one that fits parabolas is the QuadReg. What values come out of QuadReg?

Year 1990 1992 1994 1996 1998 1999

Days AQI > 100 42 20 15 25 50 61

D

t

y = ax2 + bx +ca = 1.75788b = -13.3236c = 41.0083R2 = .98824

Coefficient of Determination

The coefficient of determination, R2, describes the percent of variability in y that is explained by the model.

Values of R2 vary from 0 to 1.

The closer to 1, the better the likelihood of a good fit of the quadratic regression model (equation) to the data.

Activity contUse the model, y = 1.75788x2 – 13.3236x + 41.0083, to estimate the number of days the AQI exceeded 100 in Atlanta in each of the following years.

What is the “reliability” of the estimates?

Year 1988 1990 1992 1994 1996 1997 1998 1999 2002

Days AQI > 100 42 20 15 25 50 61

D

t

1988 and 2002 are extrapolations and of questionable reliability.

The 1997 estimate is a interpolation and therefore of better reliability.

3475 135

Activity contWe discovered that the number of days the AQI exceeded 100 in 1997 was 31 and added that to the model, how does the model change now?

How does R2 change? Why?

D

t

Before Aftery = ax2 + bx +c y = ax2 + bx +ca = 1.75788 a = 1.78324b = -13.3236 b = -13.6353c = 41.0083 c = 41.2943R2 = .98824 R2 = .98456

Year 1990 1992 1994 1996 1997 1998 1999

Days AQI > 100 42 20 15 25 31 50 61

Goes down. New value not on curve – adds variability.

Quadratic Regression ExampleThe following data from the National Health and Nutrition Examination Survey shows the percentage of American adults who are overweight or obese.

Let t represent the number of years since 1960 (t in L1 and Days in L2). Therefore, t = 1 corresponds to the year 1961. What is the model?

Year 1961 1972 1978 1991 1999

t 1 12 18 31 39

% Overweight or Obese 42 20 15 25 61

P(x) = 0.017x2 – 0.174x + 45.493

Summary and Homework

• Summary– Parabolic data can be modeled by a quadratic

regression model

• Homework– pg 457 – 461; problems 1 and 2