Question - Behavior of a Quadratic Function as x Approaches Infinity
mathquadratic function behaviorgraph behavior infinityfunction approaching infinityquadratic function characteristics
Solution:
Certainly. The question asks us to describe the behavior of the graph of the function $$ h(x) = -x^2 - 3x - 13 $$ as $$ x $$ approaches positive or negative infinity.This function $$ h(x) = -x^2 - 3x - 13 $$ is a quadratic function. It opens downwards because the coefficient of the $$ x^2 $$ term is negative. This negative coefficient means that as $$ x $$ becomes very large (positively or negatively), the value of $$ h(x) $$ will become increasingly negative, because the $$ -x^2 $$ term will dominate the value of the function for large absolute values of $$ x $$.So, as $$ x $$ approaches $$ +\infty $$, $$ h(x) $$ approaches $$ -\infty $$, indicating that the graph falls off to negative infinity.Similarly, as $$ x $$ approaches $$ -\infty $$, $$ h(x) $$ also approaches $$ -\infty $$, since the square of a large negative number is positive, and multiplying it by the negative coefficient will result in a large negative number.In summary, the graph of $$ h(x) $$ falls off to negative infinity in both directions, to the left as $$ x $$ approaches negative infinity and to the right as $$ x $$ approaches positive infinity.
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