Questions: Question 3 End Behavior of a Power Function Identify the End Behaviors for the function graphed to the right Right hand end behavior: As x → □ ,f(x) → □ Left hand end behavior: As x → □ ,f(x) → □ Question 4 Describe the long run behavior of f(x)=x^6-2x^5+2x^3-5 As x → -∞, f(x) → □ ? As x → ∞, f(x) → □ ?

Question 3
End Behavior of a Power Function
Identify the End Behaviors for the function graphed to the right

Right hand end behavior:
As x → □ ,f(x) → □

Left hand end behavior:
As x → □ ,f(x) → □

Question 4
Describe the long run behavior of f(x)=x^6-2x^5+2x^3-5
As x → -∞, f(x) → □ ?

As x → ∞, f(x) → □ ?

Transcript text: Question 3 End Behavior of a Power Function Identify the End Behaviors for the function graphed to the right Right hand end behavior: As $x \rightarrow$ $\square$ ,$f(x) \rightarrow$ $\square$ Left hand end behavior: As $x \rightarrow$ $\square$ ,$f(x) \rightarrow$ $\square$ Question 4 Describe the long run behavior of $f(x)=x^{6}-2 x^{5}+2 x^{3}-5$ As $x \rightarrow-\infty, f(x) \rightarrow$ $\square$ ? As $x \rightarrow \infty, f(x) \rightarrow$ $\square$ ?

Solution

Solution Steps

Step 1: Analyze the graph for right-hand end behavior

As x goes to positive infinity (moves to the right), the graph of f(x) goes upwards, meaning f(x) approaches positive infinity.

Step 2: Analyze the graph for left-hand end behavior

As x goes to negative infinity (moves to the left), the graph of f(x) also goes upwards, meaning f(x) approaches positive infinity.

Step 3: Determine the long run behavior of the polynomial

The dominant term in the polynomial _f(x) = x⁶ - 2x⁵ + 2x³ - 5_ is _x⁶_. Since the exponent is even and the coefficient is positive, as x approaches both positive and negative infinity, f(x) will approach positive infinity.