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Unit 5 Quadratics

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Quadratic Functions Any function that can be written in the form

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Quadratic Functions Graph forms a parabola Label the parts of the parabola or

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To find the axis of symmetry When

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Find the vertex and los

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Vertex (h,k) form of a Quadratic Standard Form:

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Parent Function

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Transformations You can tell what the graph of the quadratic will look like if the eq. is in (h,k) form

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Sketch the graph

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Identifying Important Parts on Calculator 2nd calc—then select max or min

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Completing the Square Used to go from standard form to (h,k) form or to get the equation in the form of a perfect square to solve Steps: 1.Move the constant 2.Factor out the # in front of x 2 3.Take ½ of middle term and square it 4.Write in factored form for the perfect sq. trinomial 5.Add to both sides (multiply by # in front) 6.Move constant back to get in (h,k) form

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Examples

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Example

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Solving Quadratics You can solve by graphing, factoring, square root method, and quadratic formula Solutions, roots, or zeros

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Solving by Graphing 1.Graph the parabola 2.Look for where is crosses the x-axis (where y=0) 3.May have 2 real, 1 real, or no real solutions (Show on calculator) Review finding the vertex

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Solve the following by graphing

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Solving Quadratics by Factoring 1.Factor the quadratic 2.Set each factor that contains a variable equal to zero and solve (zero product property)

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More solving by factoring

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You Try

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Writing the Quadratic Eq. Write the quadratic with the given roots of ½ and -5

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Write the quadratic with Roots of 2/3 and -2

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When solving Graphing—not always best unless you have exact answers Factoring—not every polynomial can be factored Quadratic Formula—always works Square Root method—may have to complete the square first

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Solving using Quadratic formula Must be in standard form Identify a, b, and c

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Examples

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Discriminant Used to identify the “type” of solutions you will have (without having to solve)

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If the discriminant is… A perfect square---2 rational solutions A non perfect sq—2 irrational sol. Zero—1 rational sol. Negative—2 complex sol.

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Identify the nature of the solution

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Solving Quadratics using the Sq. Rt. method Useful when you have x 2 = constant or a perfect sq. trinomial ex. (x-3) 2 =constant 1.Get the x 2 by itself 2.Take the square rt. of both sides 3.Don’t forget + or – in your answer!!!

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Examples

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Quadratic Inequalities Graphing quadratic inequalities in 2 variables: Steps: Graph the related equation Test a point not on the graph of the parabola Shade region that contains the point if it makes the inequality true or shade the other region if it does not make the inequality true Ex.

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Graphing Quadratic Inequalities

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Solving Quadratic Inequalities Solving Quadratic Inequalities in one variable: May be solved by graphing or algebraically. To solve by graphing: Steps: Put the inequality in standard form Find the zeros and sketch the graph of the related equation identify the x values for which the graph lies below the x-axis if the inequality sign is < or identify the x values for which the graph lies above the x-axis if the inequality sign is > or

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Solve by graphing Solutions:_______________________

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To solve algebraically: Steps: Solve the related equation Plot the zeros on a number line—decide whether or not the zeros are actually included in the solution set Test all regions of the number to determine other values to include in the solution set

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Solve Algebraically

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Solving Quadratic Inequalities

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Word Problems

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