Use the image to answer the question.

Question

An s-shaped curve is plotted on a coordinate plane. Both axes range from 0 to 5 in increments of 1.

Sam graphed a cube root function. What is the equation of the function Sam graphed?

(1 point)
Responses

y=x+1−−−−−√3−3
y equals root index 3 Start Root x plus 1 End Root minus 3

y=2x−1−−−−−√3+3
y equals 2 root index 3 Start Root x minus 1 End Root plus 3

y=x−1−−−−−√3+3
y equals root index 3 Start Root x minus 1 End Root plus 3

y=2x+1−−−−−√3−3

Answer 1

To identify the correct equation for the cube root function Sam graphed, we can analyze the characteristics of cube root functions.

A cube root function has the general form:

\[ y = \sqrt[3]{x} + k \]

where \( k \) is a vertical shift. The graph of \( y = \sqrt[3]{x} \) is symmetric about the origin and has an S-shaped curve, which matches the description.

From the responses given:

1. \( y = \sqrt[3]{x + 1} - 3 \)
2. \( y = 2\sqrt[3]{x - 1} + 3 \)
3. \( y = \sqrt[3]{x - 1} + 3 \)
4. \( y = 2\sqrt[3]{x + 1} - 3 \)

To find the function that describes an S-shaped curve correctly centered around the origin, we can rule out any linear transformations or shifts that do not represent such behavior.

Of the choices, the one that resembles a basic cube root function related to \( \sqrt[3]{x} \) is:

Option 3: \( y = \sqrt[3]{x - 1} + 3 \)

This function is a transformation of the standard cube root function, shifted to the right by 1 unit and up by 3 units, which retains the S-shaped curve characteristic.

Thus, the correct answer is:

y = \sqrt[3]{x - 1} + 3