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

### Parent Function

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}, wherex≠ 0.

Here are a few quadratic functions:

*y*=*x*^{2}- 5*y*=*x*^{2}- 3*x*+ 13*y*= -*x*^{2}+ 5*x*+ 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. 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,wherea≠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 = -*1*x*^{2}; (*a*= -1)*y =*1/2*x*^{2 }(*a*= 1/2)*y*= 4*x*^{2 }(*a*= 4)*y*= .25*x*^{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* = 4*x*^{2} to *y* = *x*^{2}.

*y*= 4*x*^{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* = -.25*x*^{2} to *y* = *x*^{2}.

*y*= -.25*x*^{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.