Remember that a quadratic function can be defined by equivalent expressions in different forms, which enable us to see different features of its graph. For example, these expressions define the same function:

• From factored form, we can tell that the -intercepts are  and .
• From standard form, we can tell that the -intercept is .
• From vertex form, we can tell that the vertex is .

Graph of a quadratic function on a grid. X axis from 0 to 9, by 1’s. Y axis from negative 8 to 20, by 4’s. Origin O. Graph opens upward with points 0 comma 21, 3 comma 0, and 7 comma 0. Vertex at 5 comma negative 4.

Recall that a function expressed in vertex form is written as . The values of and reveal the vertex of the graph: are the coordinates of the vertex. In this example, is 1, is 5, and is -4.

• If we have an expression in vertex form, we can rewrite it in standard form by using the distributive property and combining like terms.

  Let’s say we want to rewrite in standard form.

• If we have an expression in standard form, we can rewrite it in vertex form by completing the square.

  Let’s rewrite in vertex form.

  A perfect square would be , so we need to add 1. Adding 1, however, would change the expression. To keep the new expression equivalent to the original one, we will need to both add 1 and subtract 1.

• Let’s rewrite another expression in vertex form: .

  To make it easier to complete the square, we can use the distributive property to rewrite the expression with -2 as a factor, which gives .

  For the expression in the parentheses to be a perfect square, we need . We have 15 in the expression, so we can subtract 6 from it to get 9, and then add 6 again to keep the value of the expression unchanged. Then, we can rewrite in factored form.

  This expression is not yet in vertex form, however. To finish up, we need to apply the distributive property again so that the expression is of the form :

  When written in this form, we can see that the vertex of the graph representing is .