1. QUADRATIC FUNCTION Finding Quadratic models

2. 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

3. Avg high temp in Charlotte, NC. MonthTemp o F Mar62 April72 May80 Jun86 Jul89 Aug88 Sep82 Oct72 Nov62 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 o F. How well does your model predict the value?

4. Determine the variables Independent: Time-m represents the months of the year. model breakdown m- also should start in a sequential manner to avoid model 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.

5. 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 Temp o F Mar362 April472 May580 Jun686 Jul789 Aug888 Sep982 Oct1072 Nov1162

6. 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

7. Find a Plug in another point on the curve into the equation. Pick a point (10, 72) T(m) = a (m – 7) 2 + 89 72 = a(10 – 7) 2 + 89 72 – 89 = a(3) 2 -17 = 9a -1.89 = a

8. 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 GRAPH Should see a curve that contains the point that were listed in LIST.

9. 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

10. Check Model The actual high Temperature in Dec for Charlotte, NC is 53 o F. How well does the model predict value?

11. 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. X058101517 Y288630-1263135

12. Practice f(x) = -0.2(x +2) 2 + 7 Adjust the data X-8-5-424 Y2.85.864.2-1.2-6.8

13. Quadratic Model The median home value in thousands of dollars for Connecticut. YearMedian Home Value (Thousands $) 2004267 2005299 2006307 2007301 2008279 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.

14. Domain and Range Domain will spread out beyond the given data Range will have a maximum at the vertex and a minimum at 9

15. Solving Quadratic Equations

16. Solving Quadratic Equations C) Factoring D) Quadratic Equation

17. 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 $200,000 Find the horizontal intercepts and explain their meaning

18. 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.

19. Horizontal Intercepts When the graph touches the x-axis 0 = -8 (t – 6) 2 + 307 -307/-8 = -8 (t – 6) 2 /-8 38.375 = (t – 6) 2 (+/-)6.19= t – 6 6.19 + 6 = t -6.19 + 6 = t Represents model breakdown because median house price in 2000 and 2010 was $0. Y = 0

20. Completing the Square x 2 – 12x + 11 = 0 x 2 – 12x + 36 = -11 + 36 (x – 6) 2 = 25 x – 6 = (+/-) 5 x = 5 +6 x = -5 + 6 x = 11 and 1

21. Practice

22. Completing the Square Practice 2x 2 – 16x – 4 = 0 4a 2 + 50 = 20a

23. Factoring Equations Standard Form f(x) = x 2 + 8x + 15 Factored Form f(x) = (x + 3)(x + 5)

24. Factoring x 2 + 3x - 50 = 38 3x 2 – 5x = 28

25. Quadratic Formula

26. Practice Median home value in Gainesville, Florida, can be modeled by V(t) = -6.t 2 + 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?