Be There or Be Square

Learning Focus

Connect completing the square to the vertex, axis, and extreme value of a graph.

Big idea

Complete the square: add and subtract $\left(\frac b2\right)^2$.

$x^2+bx+c=\left(x+\frac b2\right)^2+c-\frac{b^2}{4}.$

The vertex form $(x-h)^2+k$ reveals the vertex $(h,k)$ and axis $x=h$. If $a>0$, $k$ is the minimum value.

Worked example

$y=x^2-6x+5$:

$y=(x^2-6x+9)-9+5=(x-3)^2-4.$

Vertex $(3,-4)$, axis $x=3$, minimum $-4$.

Takeaways

• Half the middle coefficient, square it, add and subtract.
• Vertex form shows the graph features directly.
• Standard form shows the $y$-intercept $c$; vertex form shows the vertex.