Essentially given a graph that depicts a cubic function whose
- Leading coefficient is positive,
- $y$ intercept is $0,4$ and the only observable $x$ intercept is $-4,0$.
Without using math beyond algebra, how could you create a equation for a graph which is guaranteed to depict that behavior but might not be the exact same graph or find the actual equation of the graph.
Best Answer
Since you said you are looking for a degree $3$ polynomial, which are often called cubic functions, I would proceed as follows:
I would take the most basic cubic function, $f(x)=x^3$ and think about how to transform it to meet the criteria you specified.
First I would translate it $4$ units to the left, which moves the $x$ intercept from $(0,0)$ to $(-4,0)$. Now the function is $f(x)=(x+4)^3$
Then I'd figure out the value of $a$ such that $f(x)=a\cdot(x+4)^3$ causes the function to have $y$ intercept at $(0,4)$
$a$ is the solution to $4=a\cdot(0+4)^3$. The solution is $a=\frac{1}{16}$.
So one function that meets your criteria is $f(x)=\frac{1}{16}\cdot(x+4)^3$, which you can expand to be $f(x)=\frac{1}{16}x^3+\frac34x^2+3x+4$