QUADRATIC FUNCTIONFinding Quadratic models

Quadratic Models

• Define Variables• Adjust data to prevent model breakdown• Draw scatter plot• Choose model type• Pick vertex and substitute into h and k• Pick another point to determine a• Write model• Check by graphing

Avg high temp in Charlotte, NC.

Month Temp oF

Mar 62

April 72

May 80

Jun 86

Jul 89

Aug 88

Sep 82

Oct 72

Nov 62

a) Find an equation for a model of these data

b) Using your model estimate the average high temperature during Dec

c) The actual avg high temp in Dec for Charlotte is 53 oF. How well does your model predict the value?

Determine the variables

• Independent:• Time-m represents the months of the year.• m- also should start in a sequential manner to avoid

model breakdownmodel breakdown(a domain value that results in an output that does not make sense or makes an equation undefined mathematically)

• Dependent:• T(m) represents the average high temperature in

degrees Fahrenheit, for each month.

Adjusted Data and Plot

• Utilizing the TI-84 enter the information into the L1 and L2

• Adjust the domain and range. x-min, x-max, y-min, and y-max

• Graph on the calculator

Month

Month Temp oF

Mar 3 62

April

4 72

May 5 80

Jun 6 86

Jul 7 89

Aug 8 88

Sep 9 82

Oct 10 72

Nov 11 62

Vertex• Determine the Vertex point.

Which point looks like the max/min?

Plug into the vertex equation:

f(x) = a(x – h) + k where h, k and a are real numbers

f(x) = a(x – 7) + 89 x-value

y-value

Find a • Plug in another point on the curve into the equation.

Pick a point (10, 72)T(m) = a (m – 7)2 + 8972 = a(10 – 7)2 + 89

72 – 89 = a(3)2

-17 = 9a-1.89 = a

Write Model• T(m) = -2.25(m – 7)2 + 89

Graph the equation on the TI-84.STAT PLOT(Y=) plot enter the equation in Y1

Enter GRAPHShould see a curve that contains the point that were listed

in LIST.

Use model to find Dec Temp• T(m) = -1.89(m – 7)2 + 89

T(m) = -1.89(12 – 7)2 + 89

T(m) =-1.89(5)2 + 89

T(m) =-1.89(25) + 89

T(m) = -47.25 + 89

T(m) = 41.75

Check Model• The actual high Temperature in Dec for Charlotte, NC is

53 oF. How well does the model predict value?

Adjusting a Model• Eyeball best fit test.• Enter the following information on the TI-84

f(x) = 4(x – 10)2 – 12

Write the equation in Y=We either need to change a, x or h.The vertex seems fine, but a needs adjustment try a

smaller value for a.

X 0 5 8 10 15 17

Y 288 63 0 -12 63 135

Practice

f(x) = -0.2(x +2)2 + 7

• Adjust the data

X -8 -5 -4 -1 2 4

Y 2.8 5.8 6 4.2 -1.2 -6.8

Quadratic ModelThe median home value in thousands of dollars for Connecticut.

Year Median Home Value (Thousands $)

2004 267

2005 299

2006 307

2007 301

2008 279

a) Find an equation for a model of these data.

b) Use your model to estimate the median home value in 2009.

c) Give a reasonable Domain and Range.

Domain and Range• Domain will spread out beyond the given data

• Range will have a maximum at the vertex and a minimum at 9

Solving Quadratic Equations•  

Solving Quadratic Equations• C) Factoring

• D) Quadratic Equation

Square Root Property• Looking at the model for the Connecticut median home

values we got: V(t) = -8(t – 6)2 + 307

Find when the median home values was $200,000Find the horizontal intercepts and explain their meaning

Median Home Value• 200 = -8(t – 6)2 + 307• 200 - 307 =-8(t – 6)2 • -107/-8=- 8(t – 6)2 /-8• 13.375 = (t – 6)2 • (+/-)3.66 = t – 6• 3.66 + 6 = t or –3.66 + 6 = t• 9.66 or 2.34• About 2010 and 2002 median home prices were 200,000.

Horizontal Intercepts

When the graph touches the x-axis0 = -8 (t – 6)2 + 307-307/-8 = -8 (t – 6)2 /-838.375 = (t – 6)2 (+/-)6.19= t – 6 6.19 + 6 = t -6.19 + 6 = tRepresents model breakdown because median house price in 2000 and 2010 was $0.

Y = 0

Completing the Square

x2 – 12x + 11 = 0

x2 – 12x + 36 = -11 + 36

(x – 6)2= 25

x – 6 = (+/-) 5

x = 5 +6

x = -5 + 6

x = 11 and 1

Practice

Completing the Square Practice• 2x2 – 16x – 4 = 0

• 4a2 + 50 = 20a

Factoring Equations• Standard Form

f(x) = x2 + 8x + 15• Factored Form

f(x) = (x + 3)(x + 5)

Factoring• x2 + 3x - 50 = 38

• 3x2 – 5x = 28

Quadratic Formula

PracticeMedian home value in Gainesville, Florida, can be modeled

by V(t) = -6.t2 + 84.4t –102.5

Where V(t) represents the median home value in thousands of dollars for Gainesville t years since 2000. In what year was the median home value $176,000?