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5-Minute Check on Activity 4-3

5-Minute Check on Activity 4-3. For the given function: y = x 2 – 6x + 9 identify the following: Vertex: Axis of Symmetry: Domain: Range: Has a max or a min: For the given function: y = -x 2 + 16 identify the following: Vertex: Axis of Symmetry: Domain: Range:

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5-Minute Check on Activity 4-3

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  1. 5-Minute Check on Activity 4-3 For the given function: y = x2 – 6x + 9 identify the following: Vertex: Axis of Symmetry: Domain: Range: Has a max or a min: For the given function: y = -x2 + 16 identify the following: Vertex: Axis of Symmetry: Domain: Range: Has a max or a min: (-b / [2a], -[b2 – 4ac] / [4a]) = (3, 0) x = -b / (2a) = 3 so x = 3 {x | x  Real #’s} {y | y  0 (y coordinate of vertex)} a > 0 so opens up and has a min (-b / [2a], -[b2 – 4ac] / [4a]) = (0, 16) x = -b / (2a) = 0 so x = 0 {x | x  Real #’s} {y | y  16 (y coordinate of vertex)} a < 0 so opens down and has a max Click the mouse button or press the Space Bar to display the answers.

  2. Activity 4 - 4 Per Capita Personal Income

  3. Objectives • Solve quadratic equations numerically • Solve quadratic equations graphically • Determine the zeros of a function using technology

  4. Vocabulary • Quadratic equation – a second order (x2) equation in form of ax2 + bx + c = 0, a ≠ 0 • Zero of the function – is the x-value of the x-intercepts of the function

  5. Activity According to statistics from the US Department of Commerce, the per capita personal income (or the average annual income) of each resident of the United States from 1960 to 2000 can be modeled by the equation: P(x) = 15.1442x2 + 98.7686x + 1831.6909 What is the practical domain from the problem? x ≥ 0

  6. Activity cont P(x) = 15.1442x2 + 98.7686x + 1831.6909 Complete the table Sketch the graph with Window (x: -5, 45, y: -1000, 35000) Estimate personal income in 1989 (x = 29) Estimate when personal income would be $20K 0 5 10 15 20 25 30 35 40 1832 2704 4334 6721 9865 13766 18425 23840 30013 about 17,400 (P(29) = $17,432) when x = 32 or about 1992

  7. Quadratic Equation Standard form of a quadratic Equation: ax2 + bx + c = 0 Examples of quadratic equations: x2 + 3x – 1 = 9 2x2 – 4x + 1 = 0 6x2 = 18

  8. Quadratic Equation Solutions Using Tables: Let Y1 = quadratic part (x-side) Let Y2 = constant Create table with x as input and y’s as the output Using Graphs: • Let Y1 = quadratic part (x-side)Let Y2 = constantGraph and find the intersection (2nd Trace) point • Subtract constant from both sides of equationLet Y1 = new quadratic equation ( = 0)Graph and find the zeros (2nd Trace) which are the x-intercepts

  9. TI-83 Help Intersection Point (2nd Trace Intersection) • Left bound must be left of intersection point • Right bound must be right of intersection point • Guess should be in between LB and RB X-intercepts (2nd Trace Zeros) • Left bound must be left of x-intercept point • Right bound must be right of x-intercept point • Guess should be in between LB and RB

  10. Quadratic Equation Solutions - Table Solve x2 + 3x – 1 = 9 using tables of data x = 2

  11. y y x x Quadratic Equation Solutions - Graph Window x:-5, 5 y: 0, 10 Find intersection points: 2nd TRACE; Intersection Solve 2x2 – 4x + 3 = 2 Solve 2x2 – 4x + 1 = 0 1st point (0.293, 2) 2nd point (1.707, 2) Find x-intercepts: 2nd TRACE; Zeros 1st point (0.293, 0) 2nd point (1.707, 0) Window x:-5, 5 y: -3, 10

  12. Summary and Homework • Summary • Quadratic equation is in form of ax2 + bx + c = 0 • Solutions to quadratic equation f(x) = nNumerically: • Construct a table for y = f(x) and determine the x-values that produce n as a y-value Graphically: • Graph y1 = f(x) and y2 = n and determine points of intersection • Graph y = f(x) – n and determine the x-intercepts of the function (the zeros of the function) • Homework • page 438; problems 9-13

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