The function you are working with is:

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

Steps for Graphing this Function:

1. Identify the transformation of the cube root function:

   • The basic function $y = \sqrt[3]{x}$ has been reflected across the x-axis due to the negative sign, and it is shifted 1 unit upwards because of the $+1$ term.

2. Key Points for the Graph:

   • At $x = 0$:
     $y = -\sqrt[3]{0} + 1 = 1$ This gives the point $(0, 1)$.

   • At $x = 1$:
     $y = -\sqrt[3]{1} + 1 = -1 + 1 = 0$ This gives the point $(1, 0)$.

   • At $x = -1$:
     $y = -\sqrt[3]{-1} + 1 = 1 + 1 = 2$ This gives the point $(-1, 2)$.

3. Behavior: The cube root function is defined for all real values of $x$, so it extends in both directions without any restrictions.

4. Symmetry: The graph is symmetric about the origin due to the odd nature of the cube root function, but the transformation changes this symmetry slightly.

Based on your image, you have positioned the points on the graph correctly. The curve reflects across the x-axis and shifts up as expected.

Would you like further details on graphing functions or transformation properties?

Here are some related questions to explore:

1. How do vertical and horizontal shifts affect graphing other functions?
2. What is the effect of reflecting a function over the x-axis or y-axis?
3. Can the cube root function be used to model real-world phenomena?
4. How does the domain of the cube root function differ from the square root function?
5. What are the key differences between even and odd functions in terms of symmetry?

Tip: When graphing, always identify how transformations (shifts, reflections, stretches) modify the basic function.