Quadratic Function - Changes in the Parabola

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How the Quadratic Function Affects Parabola Shape

The St. Louis Arch is a good example of a parabola in real life.
David Liu, Getty Images

You can use quadratic functions to explore how the equation affects the shape of a parabola. Read on to learn how to make a parabola wider or narrower or how to rotate it onto its side.

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Quadratic Function - Changes in the Parabola

A parent function is a template of domain and range that extends to other members of a function family.

Some Common Traits of Quadratic Functions

  • 1 vertex
  • 1 line of symmetry
  • The highest degree (the greatest exponent) of the function is 2
  • The graph is a parabola

Parent and Offspring

The equation for the quadratic parent function is

y = x2, where x ≠ 0.

Here are a few quadratic functions:

  • y = x2 - 5
  • y = x2 - 3x + 13
  • y = -x2 + 5x + 3

The children are transformations of the parent. Some functions will shift upward or downward, open wider or more narrow, boldly rotate 180 degrees, or a combination of the above. Use this article to learn why a parabola opens wider, opens more narrow, or rotates 180 degrees.

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Change a, Change the Graph

Another form of the quadratic function is

y = ax2 + c, where a≠ 0

In the parent function, y = x2, a = 1 (because the coefficient of x is 1).

When the a is no longer 1, the parabola will open wider, open more narrow, or flip 180 degrees.

Examples of Quadratic Functions where a ≠ 1:

  • y = -1x2;  (a = -1) 
  • y = 1/2x2 (a = 1/2)
  • y = 4x2 (a = 4)
  • y = .25x2 + 1 (a = .25)

Change a, Change the Graph

  • When a is negative, the parabola flips 180°.
  • When |a| is less than 1, the parabola opens wider.
  • When |a| is greater than 1, the parabola opens more narrow.

Keep these changes in mind when comparing the following examples to the parent function.

 

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Example 1: The Parabola Flips

Compare y = -x2 to y = x2.

Because the coefficient of -x2 is -1, then a = -1. When a is negative 1 or negative anything, the parabola will flip 180 degrees.

 



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Example 2: The Parabola Opens Wider

Compare y = (1/2)x2 to y = x2.

  • y = (1/2)x2; (a = 1/2)
  • y = x2; (a = 1)

Because the absolute value of 1/2, or |1/2|, is less than 1, the graph will open wider than the graph of the parent function.

 

 



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Example 3: The Parabola Opens More Narrow

Compare y = 4x2 to y = x2.

  • y = 4x2  (a = 4)
  • y = x2; (a = 1)

Because the absolute value of 4, or |4|, is greater than 1, the graph will open more narrow than the graph of the parent function.

 

 



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Example 4: A Combination of Changes

Compare y = -.25x2 to y = x2.

  • y = -.25x2  (a = -.25)
  • y = x2; (a = 1)

Because the absolute value of -.25, or |-.25|,  is less than 1, the graph will open wider than the graph of the parent function.

Because a is negative, the parabola of  y = -.25x2 will flip 180 degrees.

Edited by Anne Marie Helmenstine, Ph.D.