Question

What is the end behaviour of the graph?

(a) As $x$ approaches infinity, $f(x)$ approaches infinity.
As $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

(b) As $x$ approaches infinity, $f(x)$ approaches negative infinity.
As $x$ approaches negative infinity, $f(x)$ approaches infinity.

(c) As $x$ approaches infinity, $f(x)$ approaches infinity.
As $x$ approaches negative infinity, $f(x)$ approaches infinity.

(d) As $x$ approaches infinity, $f(x)$ approaches negative infinity.
As $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

Answer 1

We are given the graph of a function $f(x)$ and we are asked to select the correct option from the given options. From the graph, we can observe for any values of $x$ whether it is positive or negative, the function $f(x)$ approaches negative infinity, that is, as $x$ is getting bigger the function $f(x)$ is getting smaller and smaller approaching the negative infinity. Based on our observation, we will choose the most appropriate option from the given choices. Hence, we will have the end behavior of the given graph.

Complete step by step solution:
According to the given question, we are given the graph of a function and using the graph we have to determine the end behavior of the graph and then choose the correct option from the given four options.
From the given graph we can observe that the graph of function is open towards the negative y – axis. And we can see that as the value of $x$ increases, the function $f(x)$ also increases but then after the turning point, the value of the function $f(x)$ keeps on decreasing as the value of $x$ increases. We are trying to say as the value of the $x$ approaches infinity, the value of $f(x)$ will keep on getting smaller and approaches negative infinity.
And also, before the turning point when the value of $x$ decreases the value of the function also decreases.
That is,
$x\to \infty ,f(x)\to -\infty$
and $x\to -\infty ,f(x)\to -\infty$
Therefore, option (d) is correct.

Note: The graph should be carefully observed and the correct conclusion should be made based on the nature of the function $f(x)$. Also, if we see the graph carefully it’s the graph of a parabola which is open towards the negative y – axis and so the equation of the parabola is $y=-a{{x}^{2}}$. So, the graph of the common functions should be known to have a better understanding of the questions.