In Exercises 35 and 36, sketch a graph of a polynomial
function f having the given characteristics. • The graph of f has x-intercepts of −3, 1, and 5.
• f has a local maximum when x = 1.
• f has a local minimum when x = −2 and when
x = 4. WHAT IS THE POLYNOMIAL FUNCTION?

To find the polynomial function \( f(x) \) with the given characteristics, let's break down each of the conditions.

### Given Characteristics:
1. **x-intercepts of -3, 1, and 5**: This means that \( f(x) \) has roots at these values, so the factors corresponding to these roots will be \( (x + 3) \), \( (x - 1) \), and \( (x - 5) \).

2. **Local maximum at \( x = 1 \)**: Since \( x = 1 \) is a local maximum, the multiplicity of the root at \( x = 1 \) must be even (so that the graph touches the x-axis and then turns back). This suggests the factor \( (x - 1) \) must be squared: \( (x - 1)^2 \).

3. **Local minimum at \( x = -2 \) and \( x = 4 \)**: Local minima typically correspond to factors with odd multiplicity. The factors corresponding to \( x = -2 \) and \( x = 4 \) should be linear (i.e., first-degree factors), so these points correspond to roots of multiplicity 1.

### Step-by-step construction:

- We know that the polynomial must have the following factors:
  - \( (x + 3) \) (from the root at \( x = -3 \)),
  - \( (x - 1)^2 \) (from the local maximum at \( x = 1 \)),
  - \( (x + 2) \) (from the root at \( x = -2 \), local minimum),
  - \( (x - 4) \) (from the root at \( x = 4 \), local minimum).

Therefore, the polynomial function will look like this:

\[
f(x) = k(x + 3)(x - 1)^2(x + 2)(x - 4)
\]

Where \( k \) is a constant that determines the overall vertical stretch or shrink of the graph. This constant can be determined if additional information, like the function's value at a specific point, is provided. Since that information is not given, we will leave \( k \) as an arbitrary constant for now.

### Final Answer:

The polynomial function is:

\[
f(x) = k(x + 3)(x - 1)^2(x + 2)(x - 4)
\]

Do you want more details on graphing this polynomial, or how to determine the constant \( k \)? 

Here are some related questions:
1. How would the graph of this polynomial look like?
2. What would the derivative of this polynomial look like?
3. How do we determine the value of \( k \) using additional conditions?
4. What are the end behaviors of the graph based on the polynomial’s degree?
5. How does the multiplicity of the roots affect the graph near the x-intercepts?

**Tip:** To determine the constant \( k \), use a specific point on the graph (if available), and substitute the coordinates into the equation.

In Exercises 35 and 36, sketch a graph of a polynomial function f having the given characteristics. • The graph of f has x-intercepts of −3, 1, and 5. • f has a local maximum when x = 1. • f has a local minimum when x = −2 and when x = 4. WHAT IS THE POLYNOMIAL FUNCTION?

Discover how to determine the polynomial function f(x) with x-intercepts at -3, 1, and 5, a local maximum at x = 1, and local minima at x = -2 and x = 4. This problem involves polynomial graphing concepts, factor theorem, and root multiplicity. Suitable for high school students in Grades 10-12.

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