Let's analyze each function according to the questions provided:

1. $f(x) = 2x^4 - x^3 - 6x^2 + 3x + 2$

a) End Behavior: The end behavior of a polynomial function is determined by its leading term. For $f(x) = 2x^4 - x^3 - 6x^2 + 3x + 2$, the leading term is $2x^4$. Since the degree of the polynomial is even (4), and the leading coefficient (2) is positive, we can describe the end behavior as follows:

• As $x \to +\infty$, $f(x) \to +\infty$
• As $x \to -\infty$, $f(x) \to +\infty$

This means the graph of $f(x)$ will rise on both ends.

b) Domain and Range:

• Domain: Since this is a polynomial function, it is defined for all real values of $x$. Thus, the domain is $(-\infty, \infty)$.
• Range: Finding the exact range is more complex. However, due to the even degree (4), the graph has a minimum point and rises to $+\infty$ on both ends. Therefore, the range will be $[y_{\text{min}}, \infty)$ where $y_{\text{min}}$ is the minimum value of the function.

c) Sketch of the Graph: To sketch this graph, note the end behavior, and look for critical points where the function's derivative is zero to find the turning points. Plotting these points along with the end behavior will give an approximate shape of the function.

2. $f(x) = -4x^5 - 20x^4 + 23x^3 + 42x^2 + 20x - 8$

a) End Behavior: For $f(x) = -4x^5 - 20x^4 + 23x^3 + 42x^2 + 20x - 8$, the leading term is $-4x^5$. The degree of the polynomial is odd (5), and the leading coefficient (-4) is negative. Thus, the end behavior is:

• As $x \to +\infty$, $f(x) \to -\infty$
• As $x \to -\infty$, $f(x) \to +\infty$

This means the graph of $f(x)$ falls to $-\infty$ as $x \to +\infty$ and rises to $+\infty$ as $x \to -\infty$.

b) Domain and Range:

• Domain: Like the previous function, this is also a polynomial, so the domain is $(-\infty, \infty)$.
• Range: Since the polynomial is of odd degree (5), its range will be all real numbers, $(-\infty, \infty)$, because it goes to $+\infty$ and $-\infty$ as $x$ changes.

c) Sketch of the Graph: For the sketch, use the end behavior to understand the overall shape of the function. Then, find critical points (local maxima and minima) by setting the derivative to zero, which will help in plotting key points for an accurate graph.

Would you like a more detailed explanation of any part or help with the actual sketching of these graphs?

Related Questions

1. How do you determine the end behavior for polynomial functions with different degrees and leading coefficients?
2. What are the steps to find the range of a polynomial function with an even degree?
3. How does the sign of the leading coefficient impact the graph of a polynomial function?
4. How do you identify turning points for polynomial functions?
5. How can you determine intervals of increase and decrease for a polynomial function?

Tip

When analyzing polynomial functions, always start with the leading term for insights on end behavior; it simplifies understanding how the graph behaves as $x \to \pm \infty$.