The expression y=ax²+bx+c has always the same sign as of 'a' if:
the discriminant is negative.
That means there are no real roots.
That means the graph does not cross the x-axis.
That means that the value of y never changes sign.
so, your job is to justify why that means that y and a have the same sign.
To determine the sign of the expression y = ax^2 + bx + c in relation to the coefficient 'a', we need to consider the discriminant.
The discriminant is given by the formula Δ = b^2 - 4ac, where 'Δ' represents the discriminant and 'a', 'b', and 'c' are the coefficients of the quadratic equation.
Now, there are three possible scenarios based on the value of the discriminant:
1. If Δ > 0: In this case, the discriminant is positive. It means that the quadratic equation has two distinct real roots. The parabola opens upwards (a > 0), and the expression y = ax^2 + bx + c will have different signs on the intervals between the roots.
2. If Δ = 0: When the discriminant is equal to zero, the quadratic equation has a single real root with multiplicity. The parabola is tangent to the x-axis (a > 0), and the expression y = ax^2 + bx + c will always have the same sign.
3. If Δ < 0: When the discriminant is negative, the quadratic equation has two complex conjugate roots. The parabola opens downwards (a < 0), and the expression y = ax^2 + bx + c will always have the same sign.
Therefore, the expression y = ax^2 + bx + c will always have the same sign as 'a' if the discriminant Δ is either zero or negative.