Behavior of a Quadratic Function as x Approaches Infinity
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.
