How the Quadratic Function Affects Parabola Shape
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.
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 = x^{2}, where x ≠ 0.
Here are a few quadratic functions:
- y = x^{2} - 5
- y = x^{2} - 3x + 13
- y = -x^{2} + 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.
Change a, Change the Graph
Another form of the quadratic function is
y = ax^{2} + c, where a≠ 0
In the parent function, y = x^{2}, 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 = -1x^{2}; (a = -1)
- y = 1/2x^{2 }(a = 1/2)
- y = 4x^{2 }(a = 4)
- y = .25x^{2} + 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.
Example 1: The Parabola Flips
Compare y = -x^{2} to y = x^{2}.
Because the coefficient of -x^{2 }is -1, then a = -1. When a is negative 1 or negative anything, the parabola will flip 180 degrees.
Example 2: The Parabola Opens Wider
Compare y = (1/2)x^{2} to y = x^{2}.
- y = (1/2)x^{2}; (a = 1/2)
- y = x^{2};^{ }(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.
Example 3: The Parabola Opens More Narrow
Compare y = 4x^{2} to y = x^{2}.
- y = 4x^{2} (a = 4)
- y = x^{2};^{ }(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.
Example 4: A Combination of Changes
Compare y = -.25x^{2} to y = x^{2}.
- y = -.25x^{2} (a = -.25)
- y = x^{2};^{ }(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 = -.25x^{2 }will flip 180 degrees.
Edited by Anne Marie Helmenstine, Ph.D.