Quadratic Function - Parent Function and Vertical Shifts

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. This article focuses on vertical translations. Learn why a quadratic function shifts upward or downward.

Vertical Translations: Upward and Downward

You can also look at a quadratic function in this light:

y = x² + c, x ≠ 0

When you start with the parent function, c = 0. Therefore, the vertex (the highest or lowest point of the function) is located at (0,0).

Quick Translation Rules

1. Add c, and the graph will shift up from the parent c units.
2. Subtract c, and the graph will shift down from the parent c units.

Example 1: Increase c

When 1 is added to the parent function, the graph sits 1 unit above the parent function.

The vertex of y = x² + 1 is (0,1).

Example 2: Decrease c

When 1 is subtracted from the parent function, the graph sits 1 unit below the parent function.

The vertex of y = x² - 1 is (0,-1).

Example 3: Make a Prediction

How does y = x² + 5 differ from the parent function, y = x²?

Example 3: Answer

The function, y = x² + 5 shifts 5 units upward from the parent function.

Notice that the vertex of y = x² + 5 is (0,5), while the vertex of the parent function is (0,0).