Question

determine the parent function.

\\(y = x^2\\)

\\(y = \sqrt{x}\\)

\\(y = \frac{1}{x}\\)

Explanation:

🆕 New Concept Discovered: Parent Functions and Transformations
Recognizing the basic shape of core algebraic graphs.

Step 1: Analyze the shape of the graph

The given graph starts at a specific endpoint, \( (1, 3) \), and curves gradually upward and to the right. This "half-parabola" shape lying on its side is characteristic of a square root function.

Step 2: Compare with the options

• Option 1: \( y = x^3 \) (Cubic function)

An S-shaped curve that extends infinitely in both the positive and negative directions. This does not match our graph.

• Option 2: \( y = \sqrt{x} \) (Square root function)

A curve starting at an endpoint (the origin \( (0,0) \) in its parent form) and extending to the right. The given graph is a shifted version of this parent shape:

$$ y = \sqrt{x - 1} + 3 $$

This matches the shape perfectly.

• Option 3: \( y = \frac{1}{x} \) (Rational/Reciprocal function)

A hyperbola with two separate branches and asymptotes. This does not match our graph.

Answer:

\( y = \sqrt{x} \)