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(xh)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 = ax2 + 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).

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