We can use the discriminant of the given quadratic function to determine whether its vertex lies above, below, or on the x-axis. Recall that in the Quadratic Formula b^2-4ac is the discriminant.
ax^2+bx+c=0 ⇔ x=- b±sqrt(b^2-4ac)/2aThe value of the discriminant gives us information about the number of x-intercepts. If we just want to know whether the graph of our function intersects the x-axis and not the x-intercepts themselves, we need to work with the discriminant only. Let's now identify the values of a, b, and c.
f(x)=- 3x^2 -4x+8 ⇔ f(x)= - 3x^2+( - 4)x+ 8
Recall that when a>0 the parabola opens upwards, and when a<0 the parabola opens downwards. Therefore, from the value of the discriminant and from the sign of a we can conclude where the vertex of the given function lies without graphing.
Let's start with evaluating the discriminant.
Since the discriminant is 112, the quadratic equation has two real solutions. The graph of the given function intersects the x-axis at two points. Notice also that a= - 3, which is negative, so the parabola opens downwards. It means that the vertex must lay above the x-axis.
Extra
Further information
If the discriminant is greater than zero, the equation will have two real solutions. If it is equal to zero, the equation will have one real solution. Finally, if the discriminant is less than zero, the equation will have no real solutions.
