Quadratic Function

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. • Find an equation for a model of these data • Using your model estimate the average high temperature during Dec • 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 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

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 y-value x-value

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

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.

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, butaneeds adjustment try a smaller value for a.

Practice f(x) = -0.2(x +2)2 + 7 • Adjust the data

Quadratic Model The median home value in thousands of dollars for Connecticut. • Find an equation for a model of these data. • Use your model to estimate the median home value in 2009. • 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 $200,000 Find 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.

Y = 0 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.

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

Practice Median 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?

Quadratic Function