The Parts of a Parabola: Vertex, Axis of Symmetry and Intercepts

0:00A quadratic function like y equals x squared minus 4x minus 5 has a U-shaped graph called a parabola. What are its parts?

0:11The x-intercepts are where the graph crosses the x-axis: x equals minus 1 and x equals 5.

0:19There y is 0, so they are the roots, or solutions, of x squared minus 4x minus 5 equals 0.

0:28Factoring gives x plus 1 times x minus 5 equals 0, so x is minus 1 or 5.

0:37The axis of symmetry is the vertical line halfway between the x-intercepts: minus 1 plus 5 is 4, and 4 divided by 2 is 2, so it is the line x equals 2.

0:51Fold the graph along this line and the two halves match exactly.

0:55The vertex is the turning point. It lies on the axis of symmetry, so its x-coordinate is 2.

1:03Substitute x equals 2: 2 squared minus 4 times 2 minus 5 is 4 minus 8 minus 5, which is minus 9.

1:14The y-intercept is where the graph crosses the y-axis, at x equals 0. There y is 0 minus 0 minus 5, which is minus 5.

1:26The parabola opens upward, so the vertex is its lowest point. The range, the set of all y-values, is y greater than or equal to minus 9.

1:38Two common mistakes: the y-intercept, 0, minus 5, is not the vertex, 2, minus 9. And the axis is the line x equals 2, not a point.

1:51To sum up: x-intercepts minus 1 and 5, axis x equals 2, vertex 2, minus 9, y-intercept minus 5, and range y at least minus 9.