Chapter 3 (F11)

Chapter 3 (F11) - Section 3.1 Quadratic Functions A...

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Section 3.1 – Quadratic Functions A quadratic function is one of the form f ( x ) = a x 2 + b x + c Solving quadratic functions using the Quadratic Formula: b 2 – 4ac is called the discriminant. • If b 2 – 4ac > 0 , then the equation will have 2 real solutions • If b 2 – 4ac < 0 , then the equation will have no real solutions (there will be 2 complex sol’ns) • If b 2 – 4ac = 0 , then the equation will have 1 real solution The Quadratic Formula : If a x 2 + b x + c = 0, then x = a ac b b 2 4 2 - ± - Examples: Solve using the quadratic formula: a) 3 x 2 – 7 x + 1 = 0 b) 2 x 2 – 4 x + 3 = 0 The graphs of all quadratic functions are parabolas. Parabolas will either open up or open down , and they will contain a vertex and an axis of symmetry . 1
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Quadratic Form: f ( x ) = a x 2 + b x + c The parabola opens up if a > 0, and opens down if a < 0 The vertex (maximum/minimum) is at ( 29 ) ( , 2 2 a b a b f - - The axis of symmetry is the line x = a b 2 - The y -intercept is at (0, c) The x -intercepts are found by solving f ( x ) = 0. If b 2 – 4ac > 0, then the graph will have 2 x -intercepts If b 2 – 4ac = 0, then the graph will have 1 x -intercept If b 2 – 4ac < 0, then the graph will have 0 x -intercepts Examples : Find the vertex, axis of symmetry, x - and y -intercepts, domain and range. Sketch a graph. a) f ( x ) = x 2 + 6 x + 3 2
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b) f ( x ) = –2 x 2 + 8 x – 1 c) f ( x ) = x 2 + 4 x + 5 3
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Applications Involving Maximizing or Minimizing: Quadratic functions will always have either a maximum or a minimum value. Example : A person standing close to the edge on the top of a 160-foot building throws a baseball vertically upward. The quadratic function s ( t ) = –16 t 2 + 64 t + 160 models the ball’s height above the ground, s ( t ), in feet, t seconds after it was thrown. a) After how many seconds does the ball reach its maximum height? b) What is the maximum height? c) How many seconds does it take for the ball to hit the ground? d) Calculate s (0) and describe what it means. 4
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Example : Amy has 2500 feet of fencing available to enclose a rectangular garden. Find the dimensions of the garden that will maximize the enclosed area. What is the maximum area? 5
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Section 3.2 – Polynomial Functions and Their Graphs A Polynomial Function of Degree n has the form: f ( x ) = 0 1 1 1 a x a x a x a n n n n + + + + - - (where the a n ’s are real numbers and n is an integer ≥ 0 ) When graphed, polynomial functions are always smooth and continuous. Although the graph of a polynomial may have intervals where it increases or decreases, the graph will
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Chapter 3 (F11) - Section 3.1 Quadratic Functions A...

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