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.