Let f(x)= ax^2+bx+c, a not equal to 0 with discriminant b^2-4ac of the equation f(x)=0. Discuss the relationship between the location of the vertex of y=f(x)

Question

and the sign of the discrimination.

Answer 1

The sign of the discriminant (expression under square root in quadratic formula) tells you how many solutions there are for x.

 

A solution for x means the parabola touches/crosses the x-axis. (Once for one solution, twice for two solutions and doesn't for no solutions) So think about what a parabola looks like and how that would effect the vertex.

 

If the discriminant is < 0 (negative) there are no real solutions, only imaginary because you can not take the square root of a negative.. The parabola would never touch/cross the x-axis. The vertex of the parabola would be above the x-axis for a parabola that extends upward and below the x-axis for a parabola extending downward.

 

quadratic equation:

x = b ±√(negative number) / 2a  

 

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If the discriminant = 0 there is only one solution for x. That means the parabola may only touch/cross the x-axis once. Parabolas are symmetrical, so this can only happen if the vertex is on the x-axis.

 

quadratic equation:

x = b ±√(0) / 2a  

 

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If the discriminant > 0 (positive) there are two solutions for x so the parabola crosses the x-axis twice. The vertex would be below the x-axis for a parabola extending upward and above the x-axis for a parabola extending downward.

 

quadratic equation:

x = b ±√(positive number) / 2a

 

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