The given equation is \( y = -3(x - 4)(x - 3) \), which represents a quadratic function in factored form. To analyze this function, let's expand it and rewrite it in standard form.

### Step 1: Expand the equation

First, expand the binomials:
\[
(x - 4)(x - 3) = x^2 - 3x - 4x + 12 = x^2 - 7x + 12
\]

Now, multiply the result by \(-3\):
\[
y = -3(x^2 - 7x + 12) = -3x^2 + 21x - 36
\]

So, the equation in standard form is:
\[
y = -3x^2 + 21x - 36
\]

### Step 2: Key Features of the Parabola
- **Direction**: Since the coefficient of \(x^2\) is negative (\(-3\)), the parabola opens **downwards**.
- **Vertex**: The vertex can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 21 \), so:
  \[
  x = -\frac{21}{2(-3)} = \frac{21}{6} = 3.5
  \]
  Substituting \( x = 3.5 \) back into the equation for \( y \):
  \[
  y = -3(3.5)^2 + 21(3.5) - 36 = -3(12.25) + 73.5 - 36 = -36.75 + 73.5 - 36 = 0.75
  \]
  So, the vertex is at \( (3.5, 0.75) \).

- **X-intercepts**: The x-intercepts (or roots) are \( x = 4 \) and \( x = 3 \), as given by the factored form of the equation.
- **Y-intercept**: The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:
  \[
  y = -3(0^2) + 21(0) - 36 = -36
  \]
  So, the y-intercept is \( (0, -36) \).

Would you like more details on any part of this? Here are a few related questions:

1. How do we derive the axis of symmetry for a quadratic function?
2. What are the real-life applications of parabolas in physics or engineering?
3. Can you explain the significance of the discriminant in finding the nature of the roots?
4. How does the vertex form of a quadratic equation relate to its factored and standard forms?
5. What transformations occur when a parabola's equation is modified?

**Tip**: Always check the sign of the leading coefficient in a quadratic equation to know whether the parabola opens upwards or downwards.

Explore the quadratic function y = -3(x - 4)(x - 3) and learn how to expand it into standard form, find the vertex, x- and y-intercepts, and analyze its direction. This step-by-step solution covers factoring and vertex calculations, making it suitable for students in Grades 9-12.

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