1. Properties of Quadratics Chapter 3

2. 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.

3. 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

4. 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.

5. 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.

6. 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.

7. 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