Quadratic Function - Parent Function and Vertical Shifts

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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 = 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.

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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.
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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 = x2 + 1 is (0,1).

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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 = x2 - 1 is (0,-1).

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Example 3: Make a Prediction

A teacher explaining quadratic functions to students, Cape Town, South Africa
BFG Images/Getty Images

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

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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).

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Example 4: What is the Equation of the Green Parabola?

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Example 4: Answer

Because the vertex of the green parabola is (0,-3), its equation is y = x2 - 3.