Question

which graph represents $y = \sqrt3{x}$? (there is a graph with a parabola - like curve opening upwards with vertex at the origin, and another graph partially shown. there are radio button options.)

Explanation:

Step1: Analyze the function \( y = \sqrt[3]{x} \)

The cube root function \( y = \sqrt[3]{x} \) has a domain of all real numbers (\( x \in \mathbb{R} \)) and a range of all real numbers (\( y \in \mathbb{R} \)). It passes through the origin \((0,0)\). For negative \( x \), \( y \) is negative (since the cube root of a negative number is negative), and for positive \( x \), \( y \) is positive. It is an odd function, so it is symmetric about the origin.

Step2: Analyze the given graphs

• The first graph shown is a parabola - like shape opening upwards, symmetric about the \( y \) - axis. This is the graph of a function like \( y = x^{2} \) (even function, symmetric about \( y \) - axis, domain \( \mathbb{R} \), range \( y\geq0 \)), which does not match \( y=\sqrt[3]{x} \) because \( y = \sqrt[3]{x} \) takes negative values when \( x \) is negative.
• The second graph (partially shown) seems to start from the origin and increase slowly, but we can recall the standard graph of \( y=\sqrt[3]{x} \): when \( x = - 8 \), \( y=\sqrt[3]{-8}=-2 \); when \( x=-1 \), \( y = \sqrt[3]{-1}=-1 \); when \( x = 1 \), \( y=\sqrt[3]{1}=1 \); when \( x = 8 \), \( y=\sqrt[3]{8}=2 \). The correct graph of \( y=\sqrt[3]{x} \) should pass through points like \((-8,-2)\), \((-1,-1)\), \((0,0)\), \((1,1)\), \((8,2)\) and be symmetric about the origin. Since the first graph is of an even function (symmetric about \( y \) - axis) and the cube root function is odd (symmetric about origin), the correct graph for \( y = \sqrt[3]{x} \) is not the first one. Assuming the second graph (the one that is not the upward - opening parabola) is the cube - root graph (since it should pass through the origin, take negative values for negative \( x \) and positive for positive \( x \)). But since the first graph is incorrect, and the second graph (the non - parabola one) matches the properties of \( y=\sqrt[3]{x} \) (domain \( \mathbb{R} \), range \( \mathbb{R} \), passes through origin, symmetric about origin).

Answer:

The graph that is not the upward - opening parabola (the second graph shown, likely the one with the curve passing through the origin and having negative \( y \) for negative \( x \)) represents \( y=\sqrt[3]{x} \). If we consider the two graphs, the first is a parabola (\( y = x^{2} \) - type) and the second is the cube - root graph. So the correct graph is the non - parabola graph (the one below the first graph in the given image).