https://file.aiquickdraw.com/mathbot/file/172739030029973umlo8y.jpg

I can help you analyze this polynomial function from the image you've uploaded. Let's go step by step through the information you provided.

The polynomial function given is:

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

Now, I'll work through the different parts:

### a. Graph the function:
From the sketch, the function seems to have a behavior typical of a polynomial of degree 4. It crosses the x-axis at \(x = 4\), \(x = -1\), and \(x = \frac{1}{2}\) (based on the roots). The graph also appears to show that the function is increasing towards infinity at both extremes.

### b. Degree:
The degree of the polynomial is 4. This is because the highest power of \(x\) would come from multiplying the terms:

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

Expanding the terms would lead to \(x^4\).

### c. Sign - Leading coefficient:
The leading coefficient is positive. This is because the highest degree term \(x^4\) comes from multiplying the \(2x\) in the \((2x-1)\) factor, which is positive. Therefore, the graph opens upwards.

### d. Type of Function:
Since the degree is 4 (an even number), this is an **even-degree** polynomial.

### e. Quadrants:
The graph appears to cross the y-axis at \(y = 4\) and seems to lie mainly in quadrants II (Q2) and I (Q1).

### f. Roots/zeros:
The roots of the function occur where the function equals zero:
- \(x = 4\)
- \(x = -1\) (with multiplicity 2, meaning the graph will touch the x-axis but not cross it)
- \(x = \frac{1}{2}\)

### g. Y-intercepts:
The y-intercept occurs when \(x = 0\). Plugging \(x = 0\) into the equation:

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

Thus, the y-intercept is \(y = 4\).

### h. Domain:
The domain of any polynomial function is all real numbers:
\[
\text{Domain: } x \in \mathbb{R}
\]

### i. Range:
Since this is a degree 4 polynomial with a positive leading coefficient, the graph tends to infinity as \(x\) goes to both positive and negative infinity. Therefore, the range is also all real numbers:
\[
\text{Range: } y \in \mathbb{R}
\]

### j. End Behaviors:
As \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to \infty\), because the graph opens upwards.

### k. Local Maximums and Minimums:
This part involves identifying turning points from the graph. It seems from the sketch that there may be a local minimum near \(x = 1\) and a local maximum near \(x = -2\), but this is not confirmed in the graph. Calculating derivatives would help find these points exactly.

### l. Interval of Positive & Negative of the Function:
The function is negative between the roots \(x = 4\) and \(x = \frac{1}{2}\) and positive otherwise. Based on the roots:
- Positive when \(x \in (-\infty, -1) \cup (\frac{1}{2}, 4)\)
- Negative when \(x \in (-1, \frac{1}{2})\)

### m. Interval of Increase & Decrease of the Function:
This can be determined by analyzing the first derivative, but visually, the function increases in intervals \( (-\infty, -1) \cup (1/2, \infty)\) and decreases between \( (-1, 1/2)\).

---

Would you like me to calculate any of these features in more detail, such as finding the exact turning points?

### Relative Questions:
1. How do you find the multiplicity of a root in a polynomial?
2. What impact does the leading coefficient have on the shape of the graph?
3. How can you verify the intervals of increase and decrease mathematically?
4. What determines whether a polynomial function has a maximum or minimum?
5. Why do even-degree polynomials generally open upwards or downwards?

### Tip:
Always analyze the multiplicity of the roots carefully, as it tells you whether the graph will cross or just touch the x-axis at that root.

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

Analyze and graph the polynomial function f(x) = (x - 4)(x + 1)^2(2x - 1). This problem covers roots, zeros, the degree of the polynomial, and end behavior, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation