A group of students launches a model rocket from the top of a building the students have determined the equation y= -1/2(x + 6) (x -18) to describe the height of the rocket above ground level y in metres at any ti

Question

A group of students launches a model rocket from the top of a building the students have determined the equation y= -1/2(x + 6) (x -18) to describe the height of the rocket above ground level y in metres at any ti
W x in seconds.
table with 0,2,4,6,8,10,12,14,16,18 what are the heights above ground y.

Use the equation ti determine zeros 
Determine the equation if the axis of symmetry 
Determine the maximum height if the rocket

GPT-4o mini · Answer

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 $a x^{2} + b x + c$ has an axis of symmetry given by: $x = - \frac{b}{2 a}$

To convert: $y = - \frac{1}{2} (x^{2} - 12 x - 108)$ This expands to $y = - \frac{1}{2} x^{2} + 6 x + 54$ .

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.