Question

guided practice
determine the end - behavior for the following function.
$f(x)=-3x^{2}+5x + 3$
as $x\to\infty,f(x)\to-\infty$, as $x\to-\infty,f(x)\to-\infty$
as $x\to\infty,f(x)\to-\infty$, as $x\to-\infty,f(x)\to\infty$
as $x\to\infty,f(x)\to\infty$, as $x\to-\infty,f(x)\to-\infty$
as $x\to\infty,f(x)\to\infty$, as $x\to-\infty,f(x)\to\infty$
look at the ends of the graph.
as $x\to\infty$ means \to the right\.
as $x\to-\infty$ means \to the left\.
as $f(x)\to\infty$ means \up\.
as $f(x)\to-\infty$ means \down\.
to the left, this graph is pointing down.
to the right, this graph is pointing down.

Explanation:

Step1: Identify the degree and leading - coefficient

The function $f(x)=-3x^{2}+5x + 3$ is a quadratic function ($n = 2$, degree is 2) with leading coefficient $a=-3$.

Step2: Determine end - behavior rules

For a polynomial function $y = a x^{n}$, when $n$ is even and $a<0$, as $x
ightarrow\infty$, $y
ightarrow-\infty$ and as $x
ightarrow-\infty$, $y
ightarrow-\infty$.

Answer:

As $x
ightarrow\infty$, $f(x)
ightarrow-\infty$, as $x
ightarrow-\infty$, $f(x)
ightarrow-\infty$