Question

state the end behavior of each function shown.
6.
as ( x \to -infty ), ( f(x) \to )

as ( x \to +infty ), ( f(x) \to )

7.
as ( x \to -infty ), ( f(x) \to )

as ( x \to +infty ), ( f(x) \to )

8.
as ( x \to -infty ), ( f(x) \to )

as ( x \to +infty ), ( f(x) \to )

each function using a graphing calculator or desmos.com. based on the graph, describe the end

Explanation:

Problem 6

Step 1: Analyze left - end behavior

The graph of function 6 is a parabola opening upwards (since the coefficient of the \(x^{2}\) term is positive, or from the shape of the graph). For a parabola \(y = ax^{2}+bx + c\) with \(a>0\), as \(x
ightarrow-\infty\), we look at the trend of the graph. As \(x\) becomes very negative, the value of \(x^{2}\) is positive and large, so \(f(x)\) will tend to \(+\infty\).

Step 2: Analyze right - end behavior

Since the parabola opens upwards, as \(x
ightarrow+\infty\), \(x^{2}\) is positive and large, so \(f(x)\) will also tend to \(+\infty\).

Step 1: Analyze left - end behavior

The graph of function 7: we look at the left - most part of the graph. As \(x
ightarrow-\infty\), the graph is going upwards, so \(f(x)
ightarrow+\infty\).

Step 2: Analyze right - end behavior

As \(x
ightarrow+\infty\), the graph is going downwards, so \(f(x)
ightarrow-\infty\).

Step 1: Analyze left - end behavior

The graph of function 8: as \(x
ightarrow-\infty\), the graph is going upwards (the function values are getting larger and larger in the positive direction), so \(f(x)
ightarrow+\infty\).

Step 2: Analyze right - end behavior

As \(x
ightarrow+\infty\), the graph is approaching the \(x\) - axis from above, so the function values are getting closer and closer to 0. So \(f(x)
ightarrow0\).

Answer:

As \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\)
As \(x
ightarrow+\infty\), \(f(x)
ightarrow+\infty\)

Problem 7