To sketch a function that meets your criteria, we can create a cubic function, which is neither linear nor quadratic. Here’s a proposed function:

Function Definition

$f(x) = (x - 5)(x + 3)(x - 2)$

Properties of the Function:

1. Intercepts:

   • X-Intercepts:
     • $f(5) = 0$ → (5, 0)
     • $f(-3) = 0$ → (-3, 0)
     • $f(2) = 0$ → (2, 0)
   • Y-Intercept:
     • $f(0) = (0 - 5)(0 + 3)(0 - 2) = -30$ → (0, -30)

2. Lattice Points:

   • The x-intercepts are at (5, 0), (-3, 0), and (2, 0).
   • The y-intercept is at (0, -30).

3. Quadrants:

   • The x-intercepts span different quadrants.
   • The y-intercept is in the fourth quadrant.
   • Portions of the graph will exist below $y = -1$ (as it reaches -30) and above $y = 1$.

Sketch

You can plot the cubic function to visualize it. Here’s a brief description of the behavior:

• The graph will start from positive values, cross the y-axis at (0, -30), rise through (2, 0), dip again and cross the x-axis at (-3, 0) and finally rise again to cross (5, 0).
• Portions of the graph will be below $y = -1$ and above $y = 1$.

Tips for Graphing:

• Use graphing software or graph paper for accurate plotting.
• Mark the intercepts clearly for better visualization.

Would you like more details on the function, or do you have any questions? Here are some related questions to consider:

1. What are the general characteristics of cubic functions?
2. How can we find more lattice points for the function?
3. What are some other types of functions that fit similar criteria?
4. How do the turning points of this function relate to its intercepts?
5. How can we transform this function to change its intercepts?