# The Transformed Parabola: Vertex Form y = a(x − h)2 + k

Enter the equation *y* = (*x* − 1)2 − 2 into your calculator and sketch it. What is the equation for the axis of symmetry? What are the coordinates of the vertex? The axis is at *x* = 1, and the vertex is at the point (1, −2). The vertex form for the parabola is *y* = *a*(*x* − *h*)2 + *k*. This equation tells you that the axis of symmetry has equation *x* = *h* and that the coordinates of the vertex are (*h*, *k*). To help you remember this, the axis of symmetry for the original parabola was *x* = 0. Set the expression inside the parentheses equal to zero and solve. Because this value is also the *x*-coordinate for the vertex, when you substitute it into the equation, the number inside the parentheses will be zero; multiplied by *a* it will still be zero, and when *k* is added the result is *k*.

Find the equation for the axis of symmetry and the coordinates of the vertex for *y* = 2(*x* + 3)2 + 2.

Axis: *x* + 3 = 0 yields *x* = −3.

Vertex: *y* = −2(-3 + 3)2 + 2 = 2. The coordinates of the vertex are (-3, 2).

The process of completing the square is used to convert equations of the form *y* = *ax*2 + *bx* + *c* to vertex form. Be aware that, in contrast to the form we use when solving equations, one side of this equation is not zero but is the variable *y* instead.

The axis of symmetry is *x* = −2, and the coordinates of the vertex are (-2, −16).