# ch2.3 - Learning Objectives for Section 2.3 Quadratic...

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Barnett/Ziegler/Byleen Finite Mathematics 11e 1 Learning Objectives for Section 2.3 Quadratic Functions The student will be able to identify and define quadratic functions, equations, and inequalities. The student will be able to identify and use properties of quadratic functions and their graphs. The student will be able to solve applications of quadratic functions. The student will be able to graph and identify properties of polynomial and rational functions.

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Barnett/Ziegler/Byleen Finite Mathematics 11e 2 Quadratic Functions If a , b , c are real numbers with a not equal to zero, then the function is a quadratic function and its graph is a parabola. 2 ( ) f x ax bx c = + +
Barnett/Ziegler/Byleen Finite Mathematics 11e 3 Vertex Form of the Quadratic Function It is convenient to convert the general form of a quadratic equation to what is known as the vertex form : 2 ( ) f x ax bx c = + + 2 ( ) ( ) f x a x h k = - +

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Barnett/Ziegler/Byleen Finite Mathematics 11e 4 Completing the Square to Find the Vertex of a Quadratic Function The example below illustrates the procedure: Consider Complete the square to find the vertex. 2 ( ) 3 6 1 f x x x = - + -
Barnett/Ziegler/Byleen Finite Mathematics 11e 5 Completing the Square to Find the Vertex of a Quadratic Function The example below illustrates the procedure: Consider Complete the square to find the vertex. 2 ( ) 3 6 1 f x x x = - + - Solution:

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Barnett/Ziegler/Byleen Finite Mathematics 11e 6 Completing the square (continued) The vertex is (1, 2) The quadratic function opens down since the coefficient of the Add 1 to complete the square inside the parentheses. Because of the -3 outside the parentheses, we have actually added -3, so we must add +3 to the outside.
Barnett/Ziegler/Byleen Finite Mathematics 11e 7 Intercepts of a Quadratic Function Find the x and y intercepts of 2 ( ) 3 6 1 f x x x = - + -

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Barnett/Ziegler/Byleen Finite Mathematics 11e 8 Intercepts of a Quadratic Function Find the x and y intercepts of x intercepts: Set f ( x ) = 0: Use the quadratic formula: x = = 2 ( ) 3 6 1 f x x x = - + - 2 0 3 6 1 x x = - + - 2 4 2 b b ac a - ± - 2 6 6 4( 3)( 1) 6 24 0.184,1.816 2( 3) 6 - ± - - - - ± = - -
Barnett/Ziegler/Byleen Finite Mathematics 11e 9 Intercepts of a Quadratic Function (continued) y intercept: Let x = 0. If x = 0, then y = -1, so (0, -1) is the y intercept. 2 ( ) 3 6 1 f x x x = - + - 1 ) 0 ( 6 ) 0 ( 3 ) 0 ( 2 - + - = f

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Barnett/Ziegler/Byleen Finite Mathematics 11e 10 Generalization If a 0, then the graph of f is a parabola. If a > 0, the graph opens upward. If a < 0, the graph opens downward. Vertex is (h , k) Axis of symmetry: x = h f ( h ) = k is the minimum if a > 0, otherwise the maximum Domain = set of all real numbers Range: if a < 0. If a > 0, the range is { } y y k 2 ( ) ( ) f x a x h k = - + For { } y y k
Barnett/Ziegler/Byleen Finite Mathematics 11e 11 Solving Quadratic Inequalities Solve the quadratic inequality -3 x 2 + 6 x -1 > 0

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