Quadratic applications ver b

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Quadratic applications ver b

  1. 1. Quadratic Applications Maximum and Minimum values of Quadratic Functions
  2. 2. h=initial height, v=initial velocity, and t = time <ul><li>Example # 1 A ball is thrown straight upward from ground level with </li></ul><ul><li>initial velocity of 32 feet per second. The formula h(t) =-16t^2+32t </li></ul><ul><li>Gives its height in feet h(t), after t seconds. </li></ul><ul><li>What is the maximum height reached by the ball? </li></ul><ul><li>When does the ball return to the ground? </li></ul>
  3. 3. Solution: Find the vertex of the parabola. (Use your graphing calculator to find the coordinates of the vertex.) The x value will be the time it takes to reach the maximum height and the y value will be the height. To find when the ball reaches the ground again, use the calculator to find the second zero or root. (The x value will be the time.) In this problem the ball has no initial height. Sometimes there will be a height and sometimes there will not be initial velocity if the object is at rest. Read the problem very carefully .
  4. 5. The maximum is the point (1,16). Using the graphing calculator you might need to round to the nearest integer at times. The maximum height is 16 after 1 second.
  5. 6. Quadratic Function Applications Area Problem <ul><li>Suppose that 60 meters of fencing is available to enclose a rectangular garden, one side of which will be against the side of a house. What dimensions of the garden will guarantee a maximum area? </li></ul><ul><li>Solution: Use the given information to write a quadratic function. Write the function in standard form so as to use the calculator to find the vertex of the corresponding parabola. </li></ul><ul><li>House </li></ul><ul><li>x x </li></ul><ul><li>60-2x Let x represent the width of the garden. </li></ul><ul><li>Then 60-2x represents the length, and the area A is given by A(x) = x(60-2x). Plug the equation into your calculator and find the vertex. The x value will represent the width, x=15 and the maximum area will be 450 square meters. </li></ul>
  6. 7. Quadratic Function Applications <ul><li>Example # 3 The sum of two numbers is 24. Find the two numbers if their product is to be a maximum. </li></ul><ul><li>Solution: Let x represent one of the numbers. Since the sum is 24, the other number is 24-x. Now let p represent the product of these numbers. </li></ul><ul><li>p(x) = x(24-x) </li></ul><ul><li>Put the equation in y = and find the maximum. The x value will be 12 and the maximum will be 144. </li></ul>
  7. 8. Topics to be covered on Module 7 Test Factoring, Vertex form, identifying quadratic equation, applications of quadratic equations, and finding quadratic best-fit equation <ul><li>1. Solve for x by factoring. 10x^2-13x-3=0. </li></ul><ul><li>2. Find the vertex of the parabola. F(x) = 2(x+3)^2 + 5 </li></ul><ul><li>3. Graphing Calculator skills Quadratic Regressions </li></ul>
  8. 9. Module 8 Complex Numbers <ul><li>Complex numbers are a set of numbers that are not real numbers. They are numbers that satisfy equations like </li></ul>DEFINITION OF i And
  9. 10. Example # 1 = Example # 2 = = = Example # 3 = = = =
  10. 11. Solving Quadratic Equations by Factoring <ul><li>Example # 1 </li></ul>=0 Now set each set of ( ) equal to zero and solve. x=5 and x=5 Example # 2

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