Final answer:
The graph of the function f(x) = x^3 + x^2 - x - 2 can make a maximum of 2 turns, while the graph of f(x) = x^4 - 4x^3 + 2x^2 + x + 4 can make up to 3 turns.
Step-by-step explanation:
The maximum number of turns a graph of a function can make is related to the function's degree, which is the highest power of x in the polynomial. For the polynomial f(x) = x^3 + x^2 - x - 2, the degree is 3, meaning the maximum number of turns it can have is 2. This is because a polynomial of degree n can have at most n-1 turning points.
For the second function, f(x) = x^4 - 4x^3 + 2x^2 + x + 4, the degree is 4, so the maximum number of turns the graph could make is 3, since it's one less than the degree.
When graphing, it is important to label the graph with the function f(x) and the variable x. The x and y axes must be scaled appropriately to capture the maximum x and y values, making it easy to identify intercepts, maximums, and minimums. For a constant function like f(x) = 20, where x is a real number and 0 ≤ x ≤ 20, the graph is a horizontal line between x = 0 and x = 20, with no turns since the function is of degree 0.