For the quadratic function $y=ax^2+bx+c$, we have already seen that the vertex has $x$-coordinate $-\dfrac{b}{2a}$.

$a>0$: the minimum value of $y$ occurs at $x=-\dfrac{b}{2a}$ $a<0$: the maximum value of $y$ occurs at $x=-\dfrac{b}{2a}$

The process of finding the maximum or minimum value of a functions is called optimisation.

For the quadratic function $y=ax^2+bx+c$, we have already seen that the vertex has $x$-coordinate $-\dfrac{b}{2a}$.

$a>0$: the minimum value of $y$ occurs at $x=-\dfrac{b}{2a}$ $a<0$: the maximum value of $y$ occurs at $x=-\dfrac{b}{2a}$

### Example 1

Find the maximum or minimum value of $y=x^2+6x-1$ and the corresponding value of $x$. ### Example 2

Find the maximum or minimum value of $y=-2x^2+8x+1$ and the corresponding value of $x$. ### Example 3

The profit in selling $x$ computers per day, is given $P=-3x^2 + 120x - 400$ dollars. Find the maximum profit per day.

For the quadratic function $y=ax^2+bx+c$, we have already seen that the vertex has $x$-coordinate $-\dfrac{b}{2a}$.

$a>0$: the minimum value of $y$ occurs at $x=-\dfrac{b}{2a}$ $a<0$: the maximum value of $y$ occurs at $x=-\dfrac{b}{2a}$

Since $a=1 \gt 0$, the shape of the graph is concave down and it has the minimum value.

The minimum value occurs at:

\( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{6}{2 \times 1} \\ &= -3 \\ y &= (-3)^2 + 2 \times (-3) -1 \\ &= 2 \\ \end{align} \)

So the minimum value of $y$ is $2$, occurring at $x=-3$.

The minimum value occurs at:

\( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{6}{2 \times 1} \\ &= -3 \\ y &= (-3)^2 + 2 \times (-3) -1 \\ &= 2 \\ \end{align} \)

So the minimum value of $y$ is $2$, occurring at $x=-3$.

Since $a=-2 \lt 0$, the shape of the graph is concave up and it has the maximum value.

The maximum value occurs at:

\( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{8}{2 \times (-2)} \\ &= 2 \\ y &= -2 \times 2^2 + 8 \times 2 + 1 \\ &= 9 \\ \end{align} \)

So the minimum value of $y$ is $9$, occurring at $x=2$.

The maximum value occurs at:

\( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{8}{2 \times (-2)} \\ &= 2 \\ y &= -2 \times 2^2 + 8 \times 2 + 1 \\ &= 9 \\ \end{align} \)

So the minimum value of $y$ is $9$, occurring at $x=2$.

Since $a=-3 \lt 0$, the shape of the graph is concave up and it has the maximum value.

The maximum value occurs at:

\( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{120}{2 \times (-3)} \\ &= 20 \\ y &= -3 \times 20^2 + 120 \times 20 - 400 \\ &= 800 \\ \end{align} \)

So the maximum profit is $800$ dollars, occurring when $20$ computers are sold per day.

The maximum value occurs at:

\( \begin{align} \displaystyle x &= -\dfrac{b}{2a} \\ &= -\dfrac{120}{2 \times (-3)} \\ &= 20 \\ y &= -3 \times 20^2 + 120 \times 20 - 400 \\ &= 800 \\ \end{align} \)

So the maximum profit is $800$ dollars, occurring when $20$ computers are sold per day.