For `-3gtxgt2`,find the range of `y= -3x^2+10x-21`

Options

A. `(-78,-13)`
B. `[-78,-13]`
C. `{-78,-13}`
D. none of these

Answer

To find the range of the function \( y = -3x^2 + 10x - 21 \) for \( -3 < x < 2 \), we will follow these steps:

### Step 1: Identify the function and its domain
The function is given as:
\[ y = -3x^2 + 10x - 21 \]
The domain is specified as \( -3 < x < 2 \).

### Step 2: Find the critical points
To find the maximum or minimum values of the quadratic function, we can use the vertex formula. The x-coordinate of the vertex for a quadratic \( ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
Here, \( a = -3 \) and \( b = 10 \).

Calculating the vertex:
\[ x = -\frac{10}{2 \times -3} = \frac{10}{6} = \frac{5}{3} \]

### Step 3: Check if the vertex is within the domain
Now we need to check if \( \frac{5}{3} \) lies within the interval \( -3 < x < 2 \):
- Since \( \frac{5}{3} \approx 1.67 \), it is indeed within the interval.

### Step 4: Calculate the value of \( y \) at the vertex
Now, we will substitute \( x = \frac{5}{3} \) back into the equation to find the corresponding \( y \)-value:
\[ y = -3\left(\frac{5}{3}\right)^2 + 10\left(\frac{5}{3}\right) - 21 \]
Calculating \( \left(\frac{5}{3}\right)^2 = \frac{25}{9} \):
\[ y = -3 \cdot \frac{25}{9} + \frac{50}{3} - 21 \]
\[ y = -\frac{75}{9} + \frac{150}{9} - \frac{189}{9} \]
\[ y = \frac{-75 + 150 - 189}{9} = \frac{-114}{9} = -12.67 \]

### Step 5: Calculate the values of \( y \) at the endpoints of the interval
Now, we need to evaluate \( y \) at the endpoints \( x = -3 \) and \( x = 2 \).

1. For \( x = -3 \):
\[ y = -3(-3)^2 + 10(-3) - 21 = -3(9) - 30 - 21 = -27 - 30 - 21 = -78 \]

2. For \( x = 2 \):
\[ y = -3(2)^2 + 10(2) - 21 = -3(4) + 20 - 21 = -12 + 20 - 21 = -13 \]

### Step 6: Determine the range
Since the function is a downward-opening parabola (as \( a < 0 \)), the maximum value occurs at the vertex and the minimum values occur at the endpoints. Therefore, the range of \( y \) for \( -3 < x < 2 \) is:
\[ (-78, -13) \]

### Final Answer
The range of \( y = -3x^2 + 10x - 21 \) for \( -3 < x < 2 \) is:
\[ (-78, -13) \]