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Properties of Quadratics Chapter 3

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Martin-Gay, Developmental Mathematics 2 Introduction of Quadratic Relationships The graph of a quadratic is called a parabola. The direction of the opening of the parabola can be determined from the sign of the 2 nd difference in the table of values If the 2 nd difference is positive then it opens upwards. If the 2 nd difference is negative then it opens downwards.

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Martin-Gay, Developmental Mathematics 3 Introduction of Quadratic Relationships The vertex of a parabola is the point where the graph changes direction. It will have the greatest y-coordinate if it opens down or the smallest y-coordinate if it opens up. The y-coordinate of the vertex corresponds to an optimal value. This can be either a minimum or Maximum value

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Martin-Gay, Developmental Mathematics 4 Introduction of Quadratic Relationships A parabola is symmetrical with respect to vertical line through its vertex. This line is called the axis of symmetry. If the coordinates of the vertex are (h, k), then the equation of the axis of symmetry is x = h.

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Martin-Gay, Developmental Mathematics 5 Introduction of Quadratic Relationships If the parabola crosses the x-axis, the x- coordinates of these points are called the zeros. The vertex is directly above or below the midpoint of the segment joining the zeros.

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Martin-Gay, Developmental Mathematics 6 Finding x-intercepts Recall that in grade 9 math, we found the x-intercept of linear equations by letting y = 0 and solving for x. The same method works for x-intercepts in quadratic equations. Note: When the quadratic equation is written in standard form, the graph is a parabola opening up (when a > 0) or down (when a < 0), where a is the coefficient of the x 2 term. The intercepts will be where the parabola crosses the x-axis.

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Martin-Gay, Developmental Mathematics 7 Finding x-intercepts Find the x-intercepts of the graph of y = 4x 2 + 11x + 6. The equation is already written in standard form, so we let y = 0, then factor the quadratic in x. 0 = 4x 2 + 11x + 6 = (4x + 3)(x + 2) We set each factor equal to 0 and solve for x. 4x + 3 = 0 or x + 2 = 0 4x = – 3 or x = – 2 x = – ¾ or x = – 2 So the x-intercepts are the points ( – ¾, 0) and ( – 2, 0). Example

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