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Quadratic Functions Chapter 7

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Vertex Form Vertex (h, k)

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Vertex Form a > 0, opens upward a < 0, opens downward the larger│a│is the narrower the parabola the closer a is to zero the wider the parabola

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Stretching the Unit Quadratic

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Reflecting Across the x-axis

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Translating Graphs Up/Down

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Translating Graphs Right/Left

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Graphing a Quadratic Function First graph vertex Find a point

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Draw axis of symmetry through vertex Reflect point over axis Graphing a Quadratic Function

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Finding a Quadratic Model Create a scattergram Select a vertex (Doesn’t have to be data point) Select non-vertex point Plug vertex in for h and k, and the nonvertex point for x and f(x)/y into a standard equation Solve for a Then substitute a into the standard equation

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Graph Quadratic Model Pick vertex –(70, 5) Pick point –(40, 9) xf(x) 1930 (30)12 1940 (40)9 1950 (50)7 1960 (60)6 1970 (70)5 1980 (80)6 1990 (90)7 2000 (100)10

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7.2 Graphing Quadratics in Standard Form

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Quadratic in Standard Form Find y-intercept (0, c) Find symmetric point Use midpoint formula of the x-coordinates of the symmetric points to find the x- coordinate of the vertex Plug x-coordinate of the vertex into equation for x

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Graphing Quadratics Y-intercept –(0, 7) Symmetry Point

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Graphing Quadratics (0, 7) (6, 7) Midpoint

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Vertex formula vertex formula x-coordinate y-coordinate

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Vertex Formula

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Maximum/Minimum For a quadratic function with vertex (h, k) If a > 0, then the parabola opens upward and the vertex is the minimum point (k minimum value) If a < 0, then the parabola opens downward and vertex is the maximum point (k maximum value)

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Maximum Value Model A person plans to use 200 feet of fencing and a side of her house to enclose a rectangular garden. What dimensions of the rectangle would give the maximum area? What is the area?

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Maximum area would be 50 x 100 = 5000

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7.3 Square Root Property

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Product/Quotient Property for Square Roots For a ≥ 0 and b ≥ 0, For a ≥ 0 and b > 0, Write radicand as product of largest perfect-square and another number Apply the product/quotient property for square roots

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Simplifying Radical Expressions No radicand can be a fraction No radicand can have perfect-square factors other than one No denominator can have a radical expression

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Examples

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Square Root Property Let k be a nonnegative constant. Then, is equivalent to

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Imaginary Numbers Imaginary unit, (i), is the number whose square is -1. Square root of negative number –If n is a positive real number,

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Complex Numbers A complex number is a number in the form Examples Imaginary number is a complex number, where a and b are real numbers and b ≠ 0

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Solving with Negative Square Roots

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7.4 Completing the Square

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Perfect Square Trinomial For perfect square trinomial in the form dividing by b by 2 and squaring the result gives c:

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Examples

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7.5 Quadratic Formula

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Quadratic Formula The solutions of a quadratic equation in the formare given by the quadratic formula:

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Determining the Number of Real- Number Solutions The discriminant isand can be used to determine the number of real solutions If the discriminant > 0, there are two real- number solutions If the discriminant = 0, there in one real- number solution If the discriminant < 0, there are two imaginary-number solutions (no real)

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Quadratic Formula

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Examples One real-number solutionTwo imaginary-number solutions

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Intersections with y = n lines/points at a certain height Note if the discriminant is < 0, then there are no intersections

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Quadratic Functions. Expanding to Standard Form A quadratic function is a function that can be written in the standard form below and where quadratic.

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