Quadratic Function - Parent Function and Vertical Shifts
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. 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 = x^{2} + 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
- Add c, and the graph will shift up from the parent c units.
- Subtract c, and the graph will shift down from the parent c units.
Example 1: Increase c
Notice: When 1 is added to the parent function, the graph sits 1 unit above the parent function.
The vertex of y = x^{2} + 1 is (0,1).
Example 2: Decrease c
Notice: When 1 is subtracted from the parent function, the graph sits 1 unit below the parent function.
The vertex of y = x^{2} - 1 is (0,-1).
Example 3: Make a Prediction
How does y = x^{2} + 5 differ from the parent function, y = x^{2}?
Example 3: Answer
The function, y = x^{2} + 5 shifts 5 units upward from the parent function.
Notice that the vertex of y = x^{2} + 5 is (0,5), while the vertex of the parent function is (0,0).
Example 4: What is the Equation of the Green Parabola?
Example 4: Answer
Because the vertex of the green parabola is (0,-3), its equation is y = x^{2} - 3.