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3.1 - 1 Polynomial Function A polynomial function of degree n, where n is a nonnegative integer, is a function defined by an expression of the form where a n, a n-1, …, a 1, and a 0 are real numbers, with a n ≠ 0.

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3.1 - 2 Quadratic Function A function is a quadratic function if where a, b, and c are real numbers, with a ≠ 0.

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3.1 - 3 Simplest Quadratic x (x)(x) – 2– 24 – 1– 11 00 11 24 2 3 2 – 2– 2 3 – 2– 2 4 – 3– 3 – 4– 4 – 3– 3 – 4– 4 4 range [0, ) domain (− , ) x y

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3.1 - 4 Simplest Quadratic Parabolas are symmetric with respect to a line. The line of symmetry is called the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola. Vertex Axis Opens up Opens down

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3.1 - 5 Applying Graphing Techniques to a Quadratic Function The graph of g (x) = ax 2 is a parabola with vertex at the origin that opens up if a is positive and down if a is negative. The width of the graph of g (x) is determined by the magnitude of a. The graph of g (x) is narrower than that of (x) = x 2 if a > 1 and is broader (wider) than that of (x) = x 2 if a < 1. By completing the square, any quadratic function can be written in the form the graph of F(x) is the same as the graph of g (x) = ax 2 translated h units horizontally (to the right if h is positive and to the left if h is negative) and translated k units vertically (up if k is positive and down if k is negative).

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3.1 - 6 Example 1 GRAPHING QUADRATIC FUNCTIONS Solution a. Graph the function. Give the domain and range. x (x)(x) – 1– 13 0– 2– 2 1– 5– 5 2– 6– 6 3– 5– 5 4– 2– 2 53 2 3 – 2– 2 – 6– 6 Domain (− , ) Range [– 6, )

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3.1 - 7 Example 1 GRAPHING QUADRATIC FUNCTIONS Solution b. Graph the function. Give the domain and range. Domain (− , ) Range (– , 0] 2 3 – 2– 2 – 6– 6

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3.1 - 8 Example 1 GRAPHING QUADRATIC FUNCTIONS Solution c. Graph the function. Give the domain and range. Domain (− , ) Range (– , 3] (4, 3) 3 – 2– 2 – 6– 6 x = 4

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3.1 - 9 Example 2 GRAPHING A PARABOLA BY COMPLETING THE SQUARE Solution Express x 2 – 6x + 7 in the form (x– h) 2 + k by completing the square. Graph by completing the square and locating the vertex. Complete the square.

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3.1 - 10 Example 2 GRAPHING A PARABOLA BY COMPLETING THE SQUARE Solution Express x 2 – 6x + 7 in the form (x– h) 2 + k by completing the square. Graph by completing the square and locating the vertex. Add and subtract 9. Regroup terms. Factor; simplify. This form shows that the vertex is (3, – 2)

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3.1 - 11 Example 2 GRAPHING A PARABOLA BY COMPLETING THE SQUARE Solution Graph by completing the square and locating the vertex. Find additional ordered pairs that satisfy the equation. Use symmetry about the axis of the parabola to find other ordered pairs. Connect to obtain the graph. Domain is (− , )Range is [–2, )

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3.1 - 12 Graph of a Quadratic Function The quadratic function defined by (x) = ax 2 + bx + c can be written as where

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3.1 - 13 Graph of a Quadratic Function The graph of has the following characteristics. 1.It is a parabola with vertex (h, k) and the vertical line x = h as axis. 2.It opens up if a > 0 and down is a < 0. 3.It is broader than the graph of y = x 2 if a 1. 4.The y-intercept is (0) = c. 5.If b 2 – 4ac > 0, the x-intercepts are If b 2 – 4ac = 0, the x-intercepts is If b 2 – 4ac < 0, there are no x-intercepts.

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3.1 - 14 Example 4 FINDING THE AXIS AND THE VERTEX OF A PARABOLA USING THE VERTEX FORMULA Solution Here a = 2, b = 4, and c = 5. The axis of the parabola is the vertical line Find the axis and vertex of the parabola having equation (x) = 2x 2 +4x + 5 using the vertex formula. The vertex is (– 1, (– 1)). Since (– 1) = 2(– 1) 2 + 4 (– 1) +5 = 3, the vertex is (– 1, 3).

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