Given the minimum point, what's this parabola's equation?

Given the minimum point, what's this parabola's equation?

Question

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

Suggestion

Given

1. the least value of a quadratic function,
2. where it occurs, and
3. the coefficient of $x^2$,

can we write the function straight down?

If we were asked to sketch the graph, where would the vertex be?

Solution

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.