Opening Up or Down: Maximum, Minimum and End Behaviour

0:00A ball's height in metres after t seconds is h equals negative 5 t squared plus 20 t plus 1. Does its graph have a maximum or a minimum?

0:12This is a quadratic function, so its graph is a parabola, a curve that opens up or down.

0:20The number in front of t squared is the leading coefficient, a. If a is positive, the parabola opens up. If negative, it opens down.

0:31The turning point is the vertex. Opening up, it is the lowest point, a minimum. Opening down, it is the highest, a maximum.

0:41Here a is negative 5, so the ball's graph opens down, and its vertex is a maximum.

0:49The vertex is at t equals negative b divided by 2 a, which is negative 20 divided by negative 10, or 2.

0:58Put in t equals 2: negative 5 times 4 is negative 20, plus 40, plus 1, which makes 21.

1:07So the vertex is at 2, 21: the ball reaches a maximum height of 21 metres after 2 seconds.

1:15End behaviour is where a graph heads at the far left and far right. Beyond the ball's flight, both ends head down.

1:25Compare y equals x squared minus 4 x minus 5. Here a is 1, which is positive, so it opens up, and both ends head up.

1:37The vertex is at x equals 4 divided by 2, or 2. Then y is 4 minus 8 minus 5, so the minimum is negative 9.

1:49A common mistake is to call the vertex of a downward parabola a minimum. Every other point is lower, so it is a maximum.

1:59So positive a opens up with a minimum, negative a opens down with a maximum, and both ends head the same way.