https://file.aiquickdraw.com/mathbot/file/17282664533295g23u0fk.jpg

The image provides the following information:

- **X-intercepts**: \( 0 \) and \( 6 \)
- **Point**: \( (-6, 2) \)

This seems to indicate that the task is to determine the equation of a quadratic function, given its x-intercepts and a point through which the graph passes.

### Step-by-step solution:

1. **General form of the quadratic equation with known x-intercepts**:
   The general form of a quadratic function when the x-intercepts are given is:
   \[
   y = a(x - x_1)(x - x_2)
   \]
   where \( x_1 \) and \( x_2 \) are the x-intercepts, and \( a \) is a constant.

2. **Substitute the x-intercepts**:
   The x-intercepts are \( 0 \) and \( 6 \), so the equation becomes:
   \[
   y = a(x - 0)(x - 6)
   \]
   which simplifies to:
   \[
   y = a x(x - 6)
   \]
   or:
   \[
   y = a(x^2 - 6x)
   \]

3. **Use the point \( (-6, 2) \) to find the value of \( a \)**:
   Since the point \( (-6, 2) \) lies on the graph, substitute \( x = -6 \) and \( y = 2 \) into the equation:
   \[
   2 = a((-6)^2 - 6(-6))
   \]
   Simplifying:
   \[
   2 = a(36 + 36)
   \]
   \[
   2 = a(72)
   \]
   Solving for \( a \):
   \[
   a = \frac{2}{72} = \frac{1}{36}
   \]

4. **Final equation**:
   Substitute \( a = \frac{1}{36} \) into the quadratic equation:
   \[
   y = \frac{1}{36}(x^2 - 6x)
   \]
   This is the equation of the quadratic function.

Would you like more details or have any questions about the steps? Here are 5 related questions for practice:

1. How would the equation change if the x-intercepts were different?
2. What happens to the shape of the graph when the value of \( a \) is positive or negative?
3. Can you find the y-intercept of this quadratic equation?
4. What is the vertex of the quadratic function?
5. How would you solve for the roots of the quadratic equation if they weren't given?

**Tip:** Always verify your equation by substituting the point provided to ensure it satisfies the function!

X-intercepts: 0, 6. Point: (-6, 2). Find the equation of the quadratic function.

Learn how to find the quadratic equation given the X-intercepts (0 and 6) and a point (-6, 2) with this detailed step-by-step explanation. Suitable for students in Grades 8-10.

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