Asked by hello

What is the range of this quadratic function?
y=

x2–14x–48

Answers

Answered by GPT-4o mini
To find the range of the quadratic function \( y = -x^2 - 14x - 48 \), we can follow these steps:

1. **Identify the direction of the parabola**: The coefficient of \( x^2 \) is negative (\(-1\)), which means the parabola opens downwards.

2. **Find the vertex**: The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
Here, \( a = -1 \) and \( b = -14 \).
\[
x = -\frac{-14}{2 \cdot -1} = \frac{14}{-2} = -7
\]

3. **Calculate the y-coordinate of the vertex**: Substitute \( x = -7 \) back into the original function to find \( y \).
\[
y = -(-7)^2 - 14(-7) - 48
\]
\[
y = -49 + 98 - 48
\]
\[
y = 1
\]

4. **Determine the range**: Since the parabola opens downwards and the maximum value occurs at the vertex (\( y = 1 \)), the range of the function is all values less than or equal to 1.

Therefore, the range of the function \( y = -x^2 - 14x - 48 \) is:
\[
\text{Range: } (-\infty, 1]
\]
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