Determine a viewing window that will provide a complete graph of the function.
-$$y = 4.5 x ^ { 3 } + 12 x ^ { 2 } - 82$$
A) $[ - 100,100 ]$ by $[ - 10,10 ]$
B) $[ - 5,5 ]$ by $[ - 200,50 ]$
C) $[ - 6,6 ]$ by $[ - 50,100 ]$
D) $[ - 2,1 ]$ by $[ - 100,50 ]$

Question

Quizplus · Accepted Answer

To determine a suitable viewing window for the given cubic function, we need to consider the behavior of the function at both extremes (large positive and negative values of $x$) and around its critical points. The function $y = 4.5x^3 + 12x^2 - 82$ has a leading term of $4.5x^3$, which suggests that as $x$ becomes very large or very small (in positive or negative directions), the value of $y$ will also become significantly large (positive or negative). The choice of $[-5,5]$ for $x$ values covers the region around the origin and the critical points that are likely to occur given the degree and coefficients of the polynomial. The $y$-range of $[-200,50]$ is broad enough to capture the significant variations in $y$ values, including the minimum and maximum points that the cubic function can achieve within the given $x$-range. This makes option B the most suitable choice for a complete graph of the function.

Determine a Viewing Window That Will Provide a Complete Graph y=4.5x3+12x2−82y = 4.5 x ^ { 3 } + 12 x ^ { 2 } - 82y=4.5x3+12x2−82

Determine a Viewing Window That Will Provide a Complete Graph $y = 4.5 x ^ { 3 } + 12 x ^ { 2 } - 82$