Lesson 26

Lesson 26 - Lesson 26 Quadratic Function: - a function is...

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Unformatted text preview: Lesson 26 Quadratic Function: - a function is quadratic if it is in any of the following forms: o () , where ( ) o () , where ( )( ), where o () - the degree of a quadratic function is _______ - the graph of a quadratic function a parabola Vertex of a parabola: - the turning point, the point where the graph of a quadratic function changes from increasing to decreasing, or vice versa - the -coordinate of the vertex is either the maximum or minimum function value - the -coordinate of the vertex is the input where the maximum or minimum function value occurs The graph of a quadratic function is a parabola that opens up or down (why can’t it open to the left or right?). If the leading coefficient , the parabola opens up; if the leading coefficient , the parabola opens down (why can’t the leading coefficient ?). The graph of a quadratic function will always be symmetric about a vertical line called the line of symmetry. Standard Equation of a Parabola with Vertical Axis: ( ) ) - the vertex is ( - when , the parabola opens _______ - when , the parabola opens _______ - symmetric about the line , where is the -coordinate of the vertex A quadratic function in polynomial form ( ) converted to the standard equation of a parabola completing the square. ( ) ⁄ ( ( ( ( ⁄ ) ) ( ( ( ( )) )) ) ( ) ( ) can be ) by This is the standard equation of a parabola with a vertical axis; the vertex is ( ) or ( ( )). Theorem for Locating the Vertex of a Parabola: - the vertex of a parabola has an -coordinate - the vertex is ( ( )) The vertex of a quadratic function can be found by writing the standard ( ) ). The equation of a parabola, , and identifying ( vertex can also be found by using the previous theorem, ( ( )). It can also be found by finding the average of the -intercepts, then finding the function value of the -coordinate. Example 1: An object is projected vertically upward from the top of a building with an initial velocity of 96 ft/sec. Its distance ( ) in feet above the ground after seconds is given by the equation () a. Find its maximum distance above the ground. b. Find the height of the building. Example 2: You have 1000 feet of fencing to construct six animal pens. Each pen is in the shape of rectangle, with the six pens together forming a large rectangle feet wide by feet long. a. Express the length in terms of the width b. Express the total enclosed area in terms of c. Find the dimensions that maximize the enclosed area. Example 3: A doorway has the shape of a parabolic arch and is 16 feet high at the center and 8 feet wide at the base. If a rectangular box 7 feet high must fit through the doorway, what is the maximum width the box can have? ...
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