graph is y=(3-x)(x+2)+6 which is the following are true?

To analyze the graph of the function \( y = (3 - x)(x + 2) + 6 \), we'll first simplify and understand its properties.

### Step 1: Expand the expression
\[
y = (3 - x)(x + 2) + 6
\]
Expanding the product:
\[
y = (3 \times x) + (3 \times 2) - (x \times x) - (x \times 2) + 6
\]
\[
y = 3x + 6 - x^2 - 2x + 6
\]
Simplify further:
\[
y = -x^2 + x + 12
\]

### Step 2: Identify key properties of the quadratic function

1. **Vertex:** The vertex of the parabola \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
   - Here, \( a = -1 \), \( b = 1 \).
   - \( x = -\frac{1}{2(-1)} = \frac{1}{2} \).

   Plug \( x = \frac{1}{2} \) into the equation to find the corresponding \( y \)-value:
   \[
   y = -\left(\frac{1}{2}\right)^2 + \frac{1}{2} + 12
   \]
   \[
   y = -\frac{1}{4} + \frac{1}{2} + 12 = -0.25 + 0.5 + 12 = 12.25
   \]
   The vertex is at \( \left(\frac{1}{2}, 12.25\right) \).

2. **Direction:** Since the coefficient of \( x^2 \) is negative (\( a = -1 \)), the parabola opens downwards.

3. **Y-intercept:** When \( x = 0 \):
   \[
   y = -0^2 + 0 + 12 = 12
   \]
   The y-intercept is \( (0, 12) \).

### Step 3: Assess the truth of possible statements

1. **The parabola opens downwards.**
   - **True.** Because the coefficient of \( x^2 \) is negative.

2. **The vertex is at \( \left(\frac{1}{2}, 12.25\right) \).**
   - **True.** As calculated above.

3. **The y-intercept is at \( (0, 12) \).**
   - **True.** This is found by substituting \( x = 0 \).

4. **The parabola has a maximum point.**
   - **True.** Since the parabola opens downward, the vertex represents a maximum.

5. **The function has no real roots.**
   - This needs further analysis:
     - The roots of the quadratic can be found by setting \( y = 0 \) and solving \( -x^2 + x + 12 = 0 \) using the quadratic formula.
     - This will give the exact nature of the roots, but we know the discriminant \( b^2 - 4ac \) here is positive (since it is a quadratic with a downward opening curve and the vertex is above the x-axis), meaning there are real roots.

   So, **False.** The function has real roots.

### Conclusion:
The true statements are:
1. The parabola opens downwards.
2. The vertex is at \( \left(\frac{1}{2}, 12.25\right) \).
3. The y-intercept is at \( (0, 12) \).
4. The parabola has a maximum point.

The false statement is:
5. The function has no real roots.

Would you like to see a detailed calculation for the roots or have any other questions?

Here are some related questions:
1. How do you find the roots of a quadratic equation?
2. What are the characteristics of a parabola that opens upwards?
3. How does the discriminant determine the nature of the roots?
4. How do you complete the square to find the vertex form of a quadratic?
5. What is the significance of the vertex in the graph of a quadratic function?

**Tip:** The vertex of a quadratic function gives you the maximum or minimum point of the graph, which is crucial for understanding the function's behavior.

Explore the graph of the function y = (3 - x)(x + 2) + 6 with detailed analysis. Understand the vertex, direction, y-intercept, and roots of the quadratic equation. Suitable for high school students studying quadratic functions.

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