To meet these conditions, let's create a sketch of a cubic function, $f(x) = x^3 - 6x$, which is neither quadratic nor linear. Here’s a breakdown of why this function fits your requirements:

1. Not quadratic or linear: It's a cubic function.
2. x-intercept(s): The function has x-intercepts at $x = 0$ and other points.
3. y-intercept: The y-intercept is at $(0, 0)$.
4. Four lattice points: Choose lattice points, such as $(2, -8), (-2, 8), (3, -9), (-3, 9)$, which are clean integer coordinates and not fractions.
5. Lattice points in different quadrants: Points $(2, -8)$ and $(-2, 8)$ lie in different quadrants.
6. Portions below $y = -1$: Parts of the function dip below $y = -1$.
7. Portions above $y = 1$: Other parts rise above $y = 1$.

Let's plot this sketch to visualize the function.

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I'll create a graph of this function.Here is a sketch of the function $f(x) = x^3 - 6x$. As you can see:

• The graph crosses both the x- and y-axes at lattice points.
• Four lattice points are shown, two in different quadrants.
• Some portions of the graph lie above $y = 1$ and others below $y = -1$.

Do you want more details or have any questions?

Here are some related questions:

1. How can we find x-intercepts and y-intercepts of other polynomial functions?
2. What are the properties of cubic functions compared to quadratic or linear functions?
3. How can we determine which quadrants lattice points lie in for a function?
4. What methods can we use to identify lattice points on a function?
5. How would adding constants to this function affect its shape?

Tip: For functions that are neither quadratic nor linear, looking at powers greater than 2 (like cubic functions) can often add complexity with additional turning points and behavior across quadrants.