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Vertex Form and Factored Form of Quadratic Equations

On this web page you can gain a deeper understanding of the relationship between a quadratic function and its graph. You learned about quadratic functions in two forms:

Vertex form     y = a(xh)2 + k
and                              
Factored form     y = a(xr1)(xr2)

You'll explore how the parameters—the variables a, h, k, r1, and r2—determine the location and shape of the parabola.

Sketch

The sketch below shows the parabola y = a(xh)2 + k. Change the values of a, h, and k with the sliders. You can use positive and negative real values. Press Start Over to return to the graph of y = x2. (Note: The caret symbol, ^, represents an exponent, as on your calculator..)

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Investigate

  1. What effect does parameter a have on the shape or location of the parabola? Discuss both the sign and magnitude of a.
  2. What effect does parameter h have on the shape or location of the parabola?
  3. What effect does parameter k have on the shape or location of the parabola?
  4. Use what you've learned to write the equation of a parabola that is narrower than y = x2, opens downward, and has vertex (2, –4). Use the sketch to check your work.

Sketch

This sketch shows the parabola y = a(xr1)(xr2). Change the values of a, r1, and r2 with the sliders. You can use positive and negative real values. Press Start Over to return to the graph of y = x2.

Sorry, this page requires a Java-compatible web browser. If you're using a recent version of your browser, be sure to check its Preferences or Options to make sure that Java content is enabled. Note: Java-compatible browsers are Internet Explorer and Opera.

Investigate

  1. What effect does parameter a have on the shape or location of the parabola? Discuss both the sign and magnitude of a.
  2. What effect does parameter r1 have on the shape or location of the parabola?
  3. What effect does parameter r2 have on the shape or location of the parabola?
  4. Use what you've learned to write the equation of a parabola that is wider than y = x2, opens upward, and has x-intercepts 3 and 5. Use the sketch to check your work.