To analyze the model rocket's height given by the equation \( y = -\frac{1}{2}(x + 6)(x - 18) \), let’s tackle each part step-by-step.
1. Height Calculation at Given Time Intervals
We'll calculate the height \( y \) for each time \( x \) in the table: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18.
Equation:
\[ y = -\frac{1}{2}(x + 6)(x - 18) \]
Calculations:
-
\( x = 0 \):
\[ y = -\frac{1}{2}(0 + 6)(0 - 18) = -\frac{1}{2}(6)(-18) = 54 \] -
\( x = 2 \):
\[ y = -\frac{1}{2}(2 + 6)(2 - 18) = -\frac{1}{2}(8)(-16) = 64 \] -
\( x = 4 \):
\[ y = -\frac{1}{2}(4 + 6)(4 - 18) = -\frac{1}{2}(10)(-14) = 70 \] -
\( x = 6 \):
\[ y = -\frac{1}{2}(6 + 6)(6 - 18) = -\frac{1}{2}(12)(-12) = 72 \] -
\( x = 8 \):
\[ y = -\frac{1}{2}(8 + 6)(8 - 18) = -\frac{1}{2}(14)(-10) = 70 \] -
\( x = 10 \):
\[ y = -\frac{1}{2}(10 + 6)(10 - 18) = -\frac{1}{2}(16)(-8) = 64 \] -
\( x = 12 \):
\[ y = -\frac{1}{2}(12 + 6)(12 - 18) = -\frac{1}{2}(18)(-6) = 54 \] -
\( x = 14 \):
\[ y = -\frac{1}{2}(14 + 6)(14 - 18) = -\frac{1}{2}(20)(-4) = 40 \] -
\( x = 16 \):
\[ y = -\frac{1}{2}(16 + 6)(16 - 18) = -\frac{1}{2}(22)(-2) = 22 \] -
\( x = 18 \):
\[ y = -\frac{1}{2}(18 + 6)(18 - 18) = -\frac{1}{2}(24)(0) = 0 \]
Summary of Heights
| Time \(x\) | Height \(y\) | |------------|--------------| | 0 | 54 | | 2 | 64 | | 4 | 70 | | 6 | 72 | | 8 | 70 | | 10 | 64 | | 12 | 54 | | 14 | 40 | | 16 | 22 | | 18 | 0 |
2. Determine Zeros
Zeros of the equation occur when \( y = 0 \): \[ -\frac{1}{2}(x + 6)(x - 18) = 0 \]
Setting each factor equal to zero:
- \( x + 6 = 0 \) → \( x = -6 \)
- \( x - 18 = 0 \) → \( x = 18 \)
The zeros are at \( x = -6 \) and \( x = 18 \).
3. Determine the Axis of Symmetry
The general form for the vertex form of a quadratic function \( ax^2 + bx + c \) has an axis of symmetry given by: \[ x = -\frac{b}{2a} \]
To convert: \[ y = -\frac{1}{2}(x^2 - 12x - 108) \] This expands to \( y = -\frac{1}{2}x^2 + 6x + 54 \).
Here, \( a = -\frac{1}{2} \) and \( b = 6 \):
\[ x = -\frac{6}{2 \cdot -\frac{1}{2}} = -\frac{6}{-1} = 6 \]
The axis of symmetry is \( x = 6 \).
4. Determine Maximum Height
To find the maximum height, substitute the axis of symmetry value \( x = 6 \) into the height formula:
\[ y(6) = -\frac{1}{2}(6 + 6)(6 - 18) \]
\[ y(6) = -\frac{1}{2}(12)(-12) = 72 \]
The maximum height of the rocket is \( 72 \) meters.