Question

1. which equation is the parent function for the graph below?

options:
\\( y = |x| \\)
\\( y = x^2 \\)
\\( y = \sqrt{x} \\)
\\( y = x^3 \\)
\\( y = \sqrt3{x} \\)
clear all

Explanation:

Step1: Analyze each function's domain and shape

• \( y = |x| \): V - shaped, domain all real numbers, range \( y\geq0 \). The graph here has negative \( y \)-values, so eliminate.
• \( y = x^2 \): Parabola opening up, range \( y\geq0 \). The graph has negative \( y \)-values, eliminate.
• \( y=\sqrt{x} \): Domain \( x\geq0 \), range \( y\geq0 \). The graph has \( x<0 \), eliminate.
• \( y = x^3 \): Cubic function, symmetric about origin, passes through \((0,0)\), \((1,1)\), \((- 1,-1)\). The given graph's shape (decreasing, with a curve) matches the cubic function's behavior (for \( x<0 \), \( y = x^3 \) is decreasing as \( x \) increases towards 0, and for \( x>0 \) also has a cubic - like curve when considering transformations, but here the graph's general shape aligns with \( y = x^3 \)’s parent function shape compared to others.
• \( y=\sqrt[3]{x} \): Also a cubic - root function, but its shape (more linear - like near origin) doesn't match as well as \( y = x^3 \) for the given graph's curve.

Step2: Confirm the match

The graph has a cubic - like curve, with negative \( y \)-values for negative \( x \) (since when \( x<0 \), \( y = x^3<0 \)) and the general shape of a cubic function. The other functions are eliminated based on domain, range, or shape.

Answer:

\( y = x^3 \) (the option with \( y = x^3 \))