https://file.aiquickdraw.com/mathbot/file/1731685653211spua9yjp.jpg

The problem requires finding the equation of the parabola shown in the graph, with a given format of \( y = a(x - p)(x - q) \), where \( a \), \( p \), and \( q \) are constants.

### Steps to Determine the Equation
1. **Identify the Roots (x-intercepts)**:
   The x-intercepts are points where the parabola crosses the x-axis, specifically:
   - \( (-1, 0) \) and \( (2, 0) \).
   
   This gives us the values of \( p \) and \( q \):
   - \( p = -1 \)
   - \( q = 2 \)
   
   So, the equation becomes:
   \[
   y = a(x + 1)(x - 2)
   \]

2. **Determine the Vertex to Find \( a \)**:
   The vertex is given at \( (1, 6) \). We can use this point to find the value of \( a \).
   
   Substitute \( x = 1 \) and \( y = 6 \) into the equation:
   \[
   6 = a(1 + 1)(1 - 2)
   \]
   Simplify the equation:
   \[
   6 = a(2)(-1)
   \]
   \[
   6 = -2a
   \]
   Solving for \( a \):
   \[
   a = -3
   \]

3. **Write the Final Equation**:
   Substitute \( a = -3 \) back into the equation:
   \[
   y = -3(x + 1)(x - 2)
   \]

### Final Answer
The equation of the parabola is:
\[
y = -3(x + 1)(x - 2)
\]

Would you like more details on any step, or have any questions?

---

Here are five related questions you might find interesting:
1. How do you find the vertex of a parabola given its equation in standard form?
2. What is the significance of the coefficient \( a \) in determining the direction of a parabola?
3. How can you convert a parabola's equation from factored form to standard form?
4. How does changing the values of \( p \) and \( q \) affect the graph of a parabola?
5. What are some real-world applications of parabolic equations?

**Tip**: When working with parabolas, identifying the vertex and intercepts can make it easier to write the equation in different forms.

Write the equation of the parabola shown in the graph, in the form y = a(x - p)(x - q), where a, p, and q are integers, decimals, or simplified fractions.

Learn how to find the equation of a parabola given its intercepts and vertex. This problem demonstrates the steps to write a quadratic equation in factored form, y = a(x - p)(x - q), with a detailed solution for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation