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Properties of Quadratics

Learn about the properties of quadratics and understand how to identify the direction of a parabola, find the vertex, axis of symmetry, and x-intercepts.

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Properties of Quadratics

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  1. Chapter 3 Properties of Quadratics

  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 2nd difference in the table of values • If the 2nd difference is positive then it opens upwards. • If the 2nd difference is negative then it opens downwards.

  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

  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.

  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.

  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 x2 term. The intercepts will be where the parabola crosses the x-axis.

  7. Finding x-intercepts Example Find the x-intercepts of the graph of y = 4x2 + 11x + 6. The equation is already written in standard form, so we let y = 0, then factor the quadratic in x. 0 = 4x2 + 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).

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