y = ax^2 + bx + c vs y = |ax^2 + bx + c| - when will these graphs be identical?

Question

Consider the quadratic function y = ax^2 + bx + c, where a, b, and c are real
numbers and a ≠ 0. Describe the nature of the discriminant, b^2 - 4ac, for the graphs
of y = ax^2 + bx + c and y = |ax^2 + bx + c| to be identical

Answer 1

The graphs will be different when the calculated value is negative because the absolute value will be positive there.

If ax^2+bx+c has real roots, it will cross the x-axis at those points.  Those points are where the graph crosses from positive to negative or vice versa.

If a is positive, the graph opens upward, so only the areas not between the roots will be the same.

If a is negative, the graph opens downward, so only the area between the roots is identical.

 

The roots will be given by the quadratic formula: x = (-b ±√(b^2-4ac)) / 2a

Which root is smaller?

The denominator will always have the one with the discriminant's square root subtracted as the smaller of the two.

Why? The ± part is a square root.  No square root be a negative value.

When the denominator is negative, that division will change the sign of both roots.

When a is positive, the "-" root will be smaller.

When a is negative, the "+" root will be smaller.

 

If there are no real roots, the graph does not cross the x-axis and the answer is either everywhere or nowhere.

If a is positive, the whole graph has to be above the x-axis.  They're identical everywhere.

If a is negative, the graph is below the x-axis.  They're not identical anywhere.

 

How do you tell if the there are real roots or not?

The discriminant answers that question.  This is the expression under the radical in the quadratic formula.  If it's negative, there are no real roots.