Question

Describe the location of the vertex of the parabola relative to the x-axis. How many zeros does the polynomial have? Assume p > 0.
location of parabola number of zeros
k > 0
k = 0
k < 0

Answer 1

The parabola is represented with the equation y = (1/4p)(x - h)^2 + k.


That is the vertex form of the equation, because h and k represent the coordinates of the vertex of the parabola, h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.


So the vertex is (h, k).


To place it in the graph you just need to know the values of h and k. For example, for a positive value of p, if they k and h are positive, means the vertex is in the first quadrant.


After that the number of zeros will depend on the value of k.


Again assuming p is positive, which implies that the parabola opens upward.

k = 0 means the parabola has one zero, because the vertex is just on the x-axis.


k > 0 means the parabola has none zeros, because the vertex will be above the x-axis and it will not intersect the x - axis.

k < 0 means the parabola has two zeros, because the vertex will be below the x - axis and it will intersect the x -axis at two points.


If p is negative, the parabola opens downward and the conclusions are the opposite


k < 0 the parabola has non zeros.

k > 0 the parabola has two zeros

k = 0 the parabola has one zero (just as when p is positive)