https://file.aiquickdraw.com/mathbot/file/1727569569259ymzkiln3.jpg

The polynomial function given in the image is:

\[
f(x) = (x - 4)^2 (x + 1)^2 (2x - 1)
\]

Let's analyze its characteristics step-by-step:

### a. Graph the Function
To sketch the graph, we need to analyze its behavior, roots, and end behavior. I will provide a detailed explanation, but for graphing, I recommend using graphing software or a calculator for an accurate visual.

### b. Degree
The degree of the polynomial is determined by adding the powers of each factor:

\[
(x - 4)^2 \Rightarrow \text{Degree } 2
\]
\[
(x + 1)^2 \Rightarrow \text{Degree } 2
\]
\[
(2x - 1) \Rightarrow \text{Degree } 1
\]

Adding these, the degree is:

\[
2 + 2 + 1 = 5
\]

### c. Sign - Leading Coefficient
The leading coefficient is determined by multiplying the leading terms of each factor. The leading term of each factor is:

\[
(x - 4)^2 \Rightarrow x^2
\]
\[
(x + 1)^2 \Rightarrow x^2
\]
\[
(2x - 1) \Rightarrow 2x
\]

Thus, the leading term of \( f(x) \) is:

\[
x^2 \cdot x^2 \cdot 2x = 2x^5
\]

The leading coefficient is **2**, which is positive.

### d. Type of Function
Since the polynomial is of odd degree (5), it is classified as an **odd polynomial function**.

### e. Quadrants
- As \( x \) approaches \( +\infty \), \( f(x) \) will go to \( +\infty \).
- As \( x \) approaches \( -\infty \), \( f(x) \) will go to \( -\infty \).

This means that the function will pass through the first and third quadrants as the end behavior indicates.

### f. Roots/Zeros
Setting \( f(x) = 0 \):

1. \( (x - 4)^2 = 0 \Rightarrow x = 4 \) (multiplicity 2)
2. \( (x + 1)^2 = 0 \Rightarrow x = -1 \) (multiplicity 2)
3. \( (2x - 1) = 0 \Rightarrow x = \frac{1}{2} \) (multiplicity 1)

The roots are: \( x = 4 \), \( x = -1 \), and \( x = \frac{1}{2} \).

### g. Y-intercepts
Setting \( x = 0 \):

\[
f(0) = (0 - 4)^2 (0 + 1)^2 (2 \cdot 0 - 1) = 16 \cdot 1 \cdot (-1) = -16
\]

Thus, the y-intercept is \( (0, -16) \).

### h. Domain
The domain of the polynomial is all real numbers:

\[
(-\infty, \infty)
\]

### i. Range
Given the odd degree and positive leading coefficient, the range is all real numbers:

\[
(-\infty, \infty)
\]

### j. End Behavior
- As \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \).
- As \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \).

### k. Local Maximums and Minimums
This requires calculating the first derivative \( f'(x) \) and setting it to zero to find critical points. Analysis of \( f''(x) \) (second derivative) would then determine local maximums and minimums.

### l. Interval of Positive & Negative of the Function
This can be determined by testing intervals around the roots.

### m. Interval of Increase & Decrease of the Function
This requires the first derivative \( f'(x) \).

Would you like me to go into more detail on any of these points, calculate derivatives, or sketch the graph? 

### Additional Questions for Practice:
1. What is the behavior of the graph at \( x = 4 \) and \( x = -1 \) considering their multiplicities?
2. How does the y-intercept help in graphing the function accurately?
3. What does the positive leading coefficient indicate about the direction of the graph as \( x \) approaches infinity?
4. How do the roots' multiplicities affect the shape of the graph near these points?
5. Can you determine the symmetry of the function based on its characteristics?

**Tip**: Remember that the end behavior of a polynomial is mainly influenced by its leading term. For odd degree polynomials with a positive leading coefficient, the graph will start in the third quadrant and end in the first quadrant.

Graph the polynomial function f(x) = (x - 4)^2 (x + 1)^2 (2x - 1) and list its characteristics.

Learn how to graph the polynomial function f(x) = (x - 4)^2 (x + 1)^2 (2x - 1) and analyze its key characteristics, such as degree, leading coefficient, roots, and end behavior. This problem is ideal for students in Grades 10-12.

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