Which of following statements is true about the parabola whose equation In the xy-plane is `y=(2x-6)(x+1)`?
I. The line `x=2` is vertical line of symmetry.
II. The minimum value of y is -8.
III. The y-intercepts is -6.

Options

A. I and III only
B. II and III only
C. I and II only
D. I, II, and III

Answer

To determine which statements are true about the parabola given by the equation \( y = (2x - 6)(x + 1) \), we will analyze the statements one by one.

### Step 1: Expand the equation
First, we expand the equation to standard form:
\[ y = (2x - 6)(x + 1) = 2x^2 + 2x - 6x - 6 = 2x^2 - 4x - 6 \]

### Step 2: Find the vertex
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
Here, \( a = 2 \) and \( b = -4 \):
\[ x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1 \]
Now, substitute \( x = 1 \) back into the equation to find the y-coordinate of the vertex:
\[ y = 2(1)^2 - 4(1) - 6 = 2 - 4 - 6 = -8 \]
Thus, the vertex is at the point \( (1, -8) \).

### Step 3: Determine the line of symmetry
The line of symmetry for a parabola is given by the x-coordinate of the vertex. Since the vertex is at \( x = 1 \), the line of symmetry is \( x = 1 \), not \( x = 2 \). Therefore, statement I is false.

### Step 4: Minimum value of y
From our calculations, we found that the minimum value of \( y \) is \( -8 \) at the vertex \( (1, -8) \). Thus, statement II is true.

### Step 5: Find the y-intercept
To find the y-intercept, we set \( x = 0 \) in the original equation:
\[ y = (2(0) - 6)(0 + 1) = (-6)(1) = -6 \]
Thus, the y-intercept is \( -6 \). Therefore, statement III is true.

### Conclusion
- Statement I: False (the line of symmetry is \( x = 1 \))
- Statement II: True (the minimum value of \( y \) is \( -8 \))
- Statement III: True (the y-intercept is \( -6 \))

Thus, the true statements are II and III.

### Final Answer
The true statements are II and III.

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