Determine the X-coordinate of the Maximum/Minimum of the following equation.

3y = x2 - 6x + 7

To determine the x-coordinate of the maximum or minimum of the equation \(3y = x^2 - 6x + 7\), follow these steps:

### Step 1: Simplify the equation
First, divide both sides by 3 to simplify the expression:
\[
y = \frac{1}{3}(x^2 - 6x + 7)
\]
\[
y = \frac{1}{3}x^2 - 2x + \frac{7}{3}
\]
This is now a quadratic equation of the form:
\[
y = ax^2 + bx + c
\]
where \(a = \frac{1}{3}\), \(b = -2\), and \(c = \frac{7}{3}\).

### Step 2: Find the x-coordinate of the vertex
For a quadratic equation, the x-coordinate of the vertex (which is either the maximum or minimum) is given by:
\[
x = -\frac{b}{2a}
\]
Substitute the values of \(a\) and \(b\) into this formula:
\[
x = -\frac{-2}{2 \times \frac{1}{3}} = \frac{2}{\frac{2}{3}} = 3
\]

### Conclusion:
The x-coordinate of the minimum is \(x = 3\).

### Reasoning:
Since the coefficient \(a = \frac{1}{3}\) is positive, the parabola opens upwards, meaning the vertex is a minimum point.

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

Here are 5 related questions to consider:
1. What is the y-coordinate of the minimum point?
2. How do you determine whether a quadratic function has a maximum or minimum?
3. How would the solution change if the quadratic opened downwards?
4. Can you apply this method to any quadratic equation?
5. How does completing the square confirm the vertex?

**Tip:** When dealing with quadratic equations, the sign of the \(a\)-value determines whether the function opens upward (minimum) or downward (maximum).

Determine the X-coordinate of the Maximum/Minimum of the following equation: 3y = x^2 - 6x + 7

This step-by-step solution helps you find the x-coordinate of the minimum of the quadratic equation 3y = x^2 - 6x + 7. Learn key algebraic concepts such as vertex calculation, quadratic equations, and the parabola's direction. Perfect for students in Grades 9-10.

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