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Give the algebraic form of the function in factored form. Make sure to calculate k.

To determine the algebraic form of the function in factored form, we need to analyze the graph and identify the roots and the behavior at those roots. 

### Step 1: Identify the roots

From the graph, we can see the roots (zeros) of the function where it crosses the x-axis:

1. \( x = -3 \)
2. \( x = -1 \)
3. \( x = 2 \)

### Step 2: Determine the multiplicity of the roots

Based on the behavior of the graph near the roots:
- At \( x = -3 \), the graph crosses the x-axis, indicating it is a single root.
- At \( x = -1 \), the graph just touches the x-axis and turns around, indicating it is a root of even multiplicity (likely multiplicity 2).
- At \( x = 2 \), the graph crosses the x-axis again, indicating it is a single root.

### Step 3: Form the initial polynomial

Given the roots and their multiplicities, the polynomial can be expressed as:
\[ f(x) = k(x + 3)(x + 1)^2(x - 2) \]

### Step 4: Determine the value of \( k \)

We use the given point on the graph to determine \( k \). The point provided is \( (0, 75) \).

Substitute \( x = 0 \) and \( f(0) = 75 \) into the polynomial:
\[ 75 = k(0 + 3)(0 + 1)^2(0 - 2) \]
\[ 75 = k \cdot 3 \cdot 1^2 \cdot (-2) \]
\[ 75 = k \cdot 3 \cdot (-2) \]
\[ 75 = k \cdot (-6) \]
\[ k = -\frac{75}{6} \]
\[ k = -12.5 \]

### Final polynomial in factored form

Now we substitute \( k \) back into the polynomial:
\[ f(x) = -12.5(x + 3)(x + 1)^2(x - 2) \]

This is the algebraic form of the function in factored form.

Would you like more details or have any other questions?

Here are five related questions you can ask:
1. How do you determine the multiplicity of a root?
2. Can you explain the behavior of polynomial graphs at different roots?
3. What are some common methods for factoring polynomials?
4. How can we verify the correctness of the factored form?
5. What is the significance of the constant \( k \) in polynomial functions?

**Tip:** When analyzing graphs of polynomial functions, always pay close attention to the behavior at the roots to determine their multiplicity.

Discover how to find the algebraic form of a function in factored form by calculating k. This problem involves analyzing polynomial functions, identifying roots, determining multiplicities, and applying point evaluation techniques. Suitable for Grades 10-12 students and learners interested in polynomial function analysis.

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