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Translations of Parabolas

On this web page you can explore the relationship between the equation of a translated parabola and the coordinates of its vertex.

You've already learned that if you translate the graph of y = x2 horizontally h units and vertically k units then the equation of the translated parabola is y = (xh)2 + k.

You may also see this equation written as y = k + (xh)2 or yk = (xh)2.

When you translate any equation horizontally, you can think of it as replacing x in the equation with (xh). Likewise, a vertical translation replaces y with (yk).

Sketch

This sketch shows the graph of the parent function y = x2 in gray. The red parabola is its image after a translation. Drag the vertex (h, k) to translate the image. Observe how the coordinates of the vertex and the equation of the image change. Press Start Over to move the vertex back to (0, 0). (Note: The caret symbol, ^, represents an exponent, as on your calculator.)

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Investigate

  1. Drag the vertex to (2, 0). What is the equation of the red parabola now?
  2. Drag the vertex so the equation is y = (x – 0)2 + 4. What are the coordinates of the vertex now?
  3. Drag the red parabola to show a translation horizontally 2 units and vertically 4 units. What is the equation of the red parabola? What are the coordinates of the vertex?
  4. In general, what are the coordinates of the vertex of the parabola formed when the graph of y = x2 is translated horizontally h units and vertically k units?
  5. How can this relationship help you find the equation of a parabola if you are given only its graph?