If `(-2,7)` is the highest point on the graph of` y =-2x^2-4ax +lambda`, then `lambda` equals

Options

A. 31
B. 11
C. `-1`
D. `-1/3`

Explanation

To find the value of \(\lambda\) in the equation \(y = -2x^2 - 4ax + \lambda\) given that \((-2, 7)\) is the highest point on the graph, we can follow these steps:

### Step 1: Understand the properties of the vertex
Since \((-2, 7)\) is the highest point, it is the vertex of the parabola represented by the equation. For a quadratic equation in the form \(y = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula \(x = -\frac{b}{2a}\).

### Step 2: Identify coefficients
In our equation, we have:
- \(a = -2\)
- \(b = -4a\)
- \(c = \lambda\)

### Step 3: Substitute the x-coordinate of the vertex
We know that the x-coordinate of the vertex is \(-2\):
\[ -2 = -\frac{-4a}{2 \cdot -2} \]
This simplifies to:
\[ -2 = \frac{4a}{-4} \implies -2 = -a \implies a = 2 \]

### Step 4: Substitute \(a\) back into the equation
Now that we have \(a = 2\), we can substitute this value into the equation:
\[ y = -2x^2 - 4(2)x + \lambda \]
This simplifies to:
\[ y = -2x^2 - 8x + \lambda \]

### Step 5: Use the point \((-2, 7)\) to find \(\lambda\)
Now we substitute \(x = -2\) and \(y = 7\) into the equation:
\[ 7 = -2(-2)^2 - 8(-2) + \lambda \]
Calculating the left side:
\[ 7 = -2(4) + 16 + \lambda \]
\[ 7 = -8 + 16 + \lambda \]
\[ 7 = 8 + \lambda \]

### Step 6: Solve for \(\lambda\)
Now, we can isolate \(\lambda\):
\[ \lambda = 7 - 8 = -1 \]

Thus, the value of \(\lambda\) is \(-1\).

### Final Answer:
\(\lambda = -1\)

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