Parent Functions Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Write a single equation using transformations of the parent function y=xy = \sqrt{x} that has domain (−∞,4](-\infty, 4], range [−2,∞)[-2, \infty), and passes through the point (0,0)(0, 0).

Solution

  1. 1
    Domain (−∞,4](-\infty, 4] means 4−x≥04 - x \ge 0, so use 4−x\sqrt{4 - x}. This reflects x\sqrt{x} horizontally and shifts right 4. Range [−2,∞)[-2, \infty) means subtract 22: y=4−x−2y = \sqrt{4-x} - 2.
  2. 2
    Check (0,0)(0,0): 4−0−2=2−2=0\sqrt{4-0} - 2 = 2 - 2 = 0 ✓. Domain: 4−x≥0⇒x≤44 - x \ge 0 \Rightarrow x \le 4 ✓. Range: 4−x≥0\sqrt{4-x} \ge 0 so y≥−2y \ge -2 ✓.

Answer

y=4−x−2y = \sqrt{4-x} - 2
Working backward from desired features to construct a function requires understanding how each transformation affects domain, range, and key points. Reflecting x\sqrt{x} over the yy-axis (using −x-x) reverses the domain direction, and vertical shifts move the range.

About Parent Functions

A parent function is the simplest, most basic version of a function family — the unshifted, unstretched, unreflected template. All other functions in the family are transformations of this parent. Memorizing parent function shapes allows rapid graphing of transformed versions.

Learn more about Parent Functions →

More Parent Functions Examples

Example 1 easy

Identify the parent function of [formula] and describe the transformations.

Example 2 medium

Match each equation to its parent function: (a) [formula], (b) [formula], (c) [formula], (d) [formul

Example 3 medium

Sketch the key features (domain, range, intercepts, asymptotes) of the parent function [formula].

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