QUADRATIC FUNCTION Finding 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. 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?

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.

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

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) = a(10 – 7) – 89 = a(3) = 9a = a

Write Model T(m) = -2.25(m – 7) 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.

Use model to find Dec Temp T(m) = -1.89(m – 7) T(m) = -1.89(12 – 7) T(m) =-1.89(5) T(m) =-1.89(25) + 89 T(m) = T(m) = 41.75

Check Model The actual high Temperature in Dec for Charlotte, NC is 53 o F. 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 Y

Practice f(x) = -0.2(x +2) Adjust the data X Y

Quadratic Model The median home value in thousands of dollars for Connecticut. YearMedian Home Value (Thousands $) a)Find an equation for a model of these data. b)Use your model to estimate the median home value in 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) Find when the median home values was $200,000 Find the horizontal intercepts and explain their meaning

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

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

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

Practice

Completing the Square Practice 2x 2 – 16x – 4 = 0 4a = 20a

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

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

Quadratic Formula

Practice Median home value in Gainesville, Florida, can be modeled by V(t) = -6.t t – Where V(t) represents the median home value in thousands of dollars for Gainesville t years since In what year was the median home value $176,000?