Question regarding formula for range of quadratic function

Question

While reading through my textbook i saw 2 formulas for the range of quadratic functions as follows

$$\text{When } a > 0 \text{ range is } \left[\frac{-D}{4a}, \infty\right)$$ $$\text{When } a < 0 \text{ range is } \left(-\infty, \frac{-D}{4a}\right]$$

Quite confused where these formulas come from could anyone point me in the right direction

Answer 1

Write, assuming $a\neq0$, $$ax^2+bx+c=a\left(x^2+\frac bax+\frac ca\right)=a\left(x+\frac{b}{2a}\right)^2+\frac{4ac-b^2}{4a}=aP(x)-\frac{D}{4a}$$ where $P(x)$ is a function that has range $[0,\infty)$.

When $a>0$, the range of $aP(x)$ is also $[0,\infty)$ (why?) and then the range of $aP(x)-D/4a$ is $\left[-\frac{D}{4a},\infty\right)$ (why?).

Similarly, when $a<0$, the range of $aP(x)$ is $(-\infty,0]$ (why?) and then the range of $aP(x)-D/4a$ is $\left(-\infty,-\frac{D}{4a}\right]$ (why?).

Hope this helps. :)

Answer 2

When $a>0$, it is an upward parabola so the $y$ coordinate of the vertex is the minimum value and range is $\left[−\frac{D}{4a},\infty\right)$.

When $a<0$, it is a downward parabola so $y$ coordinate of vertex is maximum value and range is $\left(−\infty,−\frac{D}{4a}\right]$.