We know that if the value of the second differences is positive, the parabola opens up (cup-shaped), and has a minimum value.
We also know that if the value of the second differences is negative, the parabola opens down (umbrella-shaped), and has a maximum value.
The maximum or minimum point of a parabola is called the vertex of the parabola.
The vertex indicates to us where the parabola is symmetrical with respect to the y-axis. (Symmetrical = same image on both sides.) This line is called the axis of symmetry of the parabola, and the vertex lies on this line. Therefore, if the coordinates of the vertex are (h,k) then the equation of the axis of symmetry is x=h
The axis of symmetry is also the perpendicular bisector of the segment joining any two points on the parabola that have the same y-coordinate. If the parabola crosses the x-axis, then the x-coordinates of these points are called the zeros, roots, or x-intercepts of the parabola. The vertex lies either directly above or directly below the midpoint of the zeros.
Therefore, the x-coordinate of the vertex can be determined by calculating the midpoint of any two points on the parabola having the same y-coordinate.
The optimal value is the highest or lowest point on the graph, and can be determined by the y-coordinate of the vertex.
Practice Questions
Page 266-68 #2bc, 3, 5, 9, 10, and 12.
