Review question

# Given the minimum point, what's this parabola's equation? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6756

## Solution

The least value of the function $x^2+px+q$ is $3$, and this occurs when $x=-2$. Find the values of $p$ and $q$.

By completing the square, we can express any quadratic polynomial in the form $a(x-r)^2+s$ for some $a$, $r$ and $s$.

From this, if $a$ is positive, the least value is $s$, and this occurs when $x=r$.

For our question, we must have that $x^2+px+q = (x+2)^2+3 = x^2+4x+7,$ and so, by comparing coefficients, we get $p=4$ and $q=7$. A sketch of the curve can be seen below.