https://file.aiquickdraw.com/mathbot/file/1729480825959ftrcmwc2.jpg

Let's analyze the graph in the image and determine its equation in factored form as requested.

### Step 1: Identify the Roots
From the graph, we can see that the curve crosses the x-axis at two points:
- \( x = -2 \)
- \( x = 2 \)

These are the **roots** of the polynomial function, so we can express the factors of the polynomial using these roots as follows:

\[
f(x) = a(x + 2)(x - 2)
\]

### Step 2: Check the Behavior at the Roots
The graph crosses the x-axis at \( x = -2 \) and \( x = 2 \), and there is no bouncing at these points, indicating that these roots are of **odd multiplicity** (likely multiplicity 1).

### Step 3: Determine the Leading Coefficient \(a\)
Next, we use another point on the graph to determine the value of \(a\). The graph clearly passes through the point \( (0, -4) \). We substitute this point into the equation to solve for \(a\):

\[
f(0) = a(0 + 2)(0 - 2) = -4
\]
\[
a(2)(-2) = -4
\]
\[
-4a = -4 \quad \Rightarrow \quad a = 1
\]

### Step 4: Write the Final Equation
Since \(a = 1\), the equation in factored form is:

\[
f(x) = (x + 2)(x - 2)
\]

Thus, the factored form of the equation representing the graph is:

\[
f(x) = (x + 2)(x - 2)
\]

This is the equation of the graph.

---

Would you like more details or have any questions about this solution?

Here are 5 related questions for further understanding:
1. How does the multiplicity of a root affect the graph's behavior at that point?
2. What is the significance of using the point \( (0, -4) \) to solve for \(a\)?
3. How would the equation change if the graph bounced off the x-axis at a root?
4. Can you explain why we only used two factors for this polynomial?
5. How would you convert this factored form equation into standard form?

**Tip**: Always check the behavior of the graph at the roots to identify their multiplicity (whether the graph crosses or touches the x-axis).

Determine an equation for the pictured graph in factored form.

Learn how to determine the equation of a polynomial function in factored form based on a graph. This problem explores the concepts of polynomial roots and leading coefficients using algebraic techniques. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation