Explain the difference between the graphs of polynomial functions with a degree of 3 that have a positive leading coefficient and the graphs of those

Result:
For polynomial functions of degree 3 with a positive leading coefficient, the graph has an upward trend as $x$ approaches positive infinity and a downward trend as $x$ approaches negative infinity. For polynomial functions of degree 3 with a negative leading coefficient, the graph has a downward trend as $x$ approaches positive infinity and an upward trend as $x$ approaches negative infinity.
Solution:
Let's consider a polynomial function with a degree of 3 and a positive leading coefficient. We can represent it as:
$f(x)=a{x}^3+b{x}^2+cx+d,$
where $a>0$.
When $x$ approaches positive infinity ($x\to +\infty$), the term $a{x}^3$ dominates the function. Since $a>0$, the function increases without bound. This means that the graph of the polynomial function will have an upward trend as $x$ goes towards positive infinity.
On the other hand, when $x$ approaches negative infinity ($x\to -\infty$), the term $a{x}^3$ still dominates the function. However, since $a>0$, the negative values of $x^3$ will be positive. Therefore, the function decreases without bound. As a result, the graph of the polynomial function will have a downward trend as $x$ approaches negative infinity.
Now let's consider a polynomial function with a degree of 3 and a negative leading coefficient. We can represent it as:
$g(x)=-a{x}^3+b{x}^2+cx+d,$
where $a>0$.
When $x$ approaches positive infinity ($x\to +\infty$), the term $-a{x}^3$ dominates the function. Since $a>0$, the function decreases without bound. Therefore, the graph of the polynomial function will have a downward trend as $x$ approaches positive infinity.
Similarly, when $x$ approaches negative infinity ($x\to -\infty$), the term $-a{x}^3$ still dominates the function. However, the negative values of $x^3$ will be negative. Thus, the function increases without bound. Consequently, the graph of the polynomial function will have an upward trend as $x$ goes towards negative infinity.

The difference between the graphs of polynomial functions with a degree of $3$ that have a positive leading coefficient and the graphs of those with a negative leading coefficient can be summarized as follows:
Polynomial functions with a positive leading coefficient ($a>0$) tend to have the following characteristics:
1. The graph starts in the bottom left quadrant and ends in the top right quadrant.
2. As $x$ approaches $-\infty$, the graph decreases without bound.
3. The graph passes through the origin (the point $(0,0)$).
4. There is at least one point of local minimum or maximum on the graph.
On the other hand, polynomial functions with a negative leading coefficient ($a<0$) exhibit the following characteristics:
1. The graph starts in the top left quadrant and ends in the bottom right quadrant.
2. As $x$ approaches $-\infty$, the graph decreases without bound.
3. The graph passes through the origin (the point $(0,0)$).
4. There is at least one point of local minimum or maximum on the graph.
In both cases, the graphs of these polynomial functions are continuous and smooth curves. However, the key distinction lies in the direction in which the graph opens and the position of the local minimum or maximum point.