How many points does #y=-2x^2+x-3# have in common with the vertex and where is the vertex in relation to the x axis?

Question

Savannah Anderson · Accepted Answer

#x_("vertex")=+1/4#

The number of points in common with the vertex and the graph is 1

This is a quadratic equation and the coefficient of #x^2# is negative. Consequently the graph is of form #nn# thus the vertex is a maximum

#color(red)("If")# the coefficient has been positive then you would have the general form of #uu#

The vertex has only one point.

#color(green)("A sort of cheat to determine x-vertex")#
Not really a cheat as it is part of the process for completing the square.

Write as: #color(darkcyan)(-2)(x^2+color(magenta)(1/(-2)) x)-3#

Note that #(color(darkcyan)(-2))xx(color(magenta)(-1/2))x=+x#

#x_("vertex")=(-1/2)xxcolor(magenta)(-1/2) = +1/4#

Avery Alexander · Answer

undefined The equation y = -2x^2 + x - 3 represents a quadratic function. The vertex of a quadratic function in the form y = ax^2 + bx + c is located at the point (h, k), where h = -b / (2a) and k = f(h), where f is the function.

For y = -2x^2 + x - 3, a = -2, b = 1, and c = -3. Plugging these values into the formulas gives h = 1/4 and k = -13/4. So, the vertex is at (1/4, -13/4).

To find the number of points the vertex has in common with the graph, we need to evaluate the discriminant of the quadratic equation, which is b^2 - 4ac.

For y = -2x^2 + x - 3, the discriminant is 1^2 - 4(-2)(-3) = 1 - 24 = -23. Since the discriminant is negative, the graph does not intersect the x-axis, so the vertex does not have any points in common with the x-axis.