What is the range of this quadratic function?
y=
–
8x2–32x+40

To find the range of the quadratic function \( y = -8x^2 - 32x + 40 \), we need to follow these steps:

### 1. Identify the vertex of the parabola:
The general form of a quadratic function is \( y = ax^2 + bx + c \). In this case, \( a = -8 \), \( b = -32 \), and \( c = 40 \).

The x-coordinate of the vertex is given by:
\[
x = \frac{-b}{2a}
\]
Substituting the values of \( a \) and \( b \):
\[
x = \frac{-(-32)}{2(-8)} = \frac{32}{-16} = -2
\]

### 2. Find the y-coordinate of the vertex:
Substitute \( x = -2 \) back into the equation to find \( y \):
\[
y = -8(-2)^2 - 32(-2) + 40
\]
\[
y = -8(4) + 64 + 40
\]
\[
y = -32 + 64 + 40 = 72
\]

### 3. Determine the range:
Since the coefficient of \( x^2 \) (which is \( a = -8 \)) is negative, the parabola opens downward. This means the vertex represents the maximum point of the parabola.

Therefore, the range of the function is:
\[
(-\infty, 72]
\]

Would you like more details or have any questions?

### 5 Related Questions:
1. How do you determine if a parabola opens upward or downward?
2. What is the significance of the vertex in the range of a quadratic function?
3. How do you find the axis of symmetry in a quadratic function?
4. Can you explain how the sign of the leading coefficient affects the shape of the parabola?
5. What are other methods to find the range of a quadratic function?

### Tip:
Remember that the vertex form of a quadratic function can make it easier to determine the range and vertex of the function directly.

Learn how to find the range of the quadratic function y = -8x^2 - 32x + 40 using the vertex form and properties of parabolas. Understand why the range of a downward opening parabola extends from negative infinity to 72. Suitable for high school students studying quadratic functions.

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