The problem asks us to graph the quadratic equation $y = -x^2 + 10x - 9$ by making a table of values. We need to find the vertex and the roots of the quadratic to determine the height and distance of the water balloon launched from a toy catapult.

1. Problem Description

The problem asks us to graph the quadratic equation $y = -x^2 + 10x - 9$ by making a table of values. We need to find the vertex and the roots of the quadratic to determine the height and distance of the water balloon launched from a toy catapult.

2. Solution Steps

First, we make a table of values for x and y.

To find the vertex of the parabola, we can use the formula for the x-coordinate of the vertex:

$x = -b/(2a)$, where $a = -1$ and $b = 10$ in the equation $y = -x^2 + 10x - 9$.

$x = -10/(2*(-1)) = -10/(-2) = 5$

Now, we find the y-coordinate of the vertex by plugging $x = 5$ into the equation:

$y = -(5)^2 + 10(5) - 9 = -25 + 50 - 9 = 25 - 9 = 16$

So, the vertex is at $(5, 16)$.

To find the roots (where the balloon lands), we set $y = 0$ and solve for $x$:

$0 = -x^2 + 10x - 9$

$x^2 - 10x + 9 = 0$

We can factor this quadratic equation:

$(x - 1)(x - 9) = 0$

So, the roots are $x = 1$ and $x = 9$.

Now, let's make a table of values around the vertex and roots:

| x | y = -x^2 + 10x - 9 |

|---|---------------------|

| 0 | -0^2 + 10(0) - 9 = -9 |

| 1 | -1^2 + 10(1) - 9 = 0 |

| 2 | -2^2 + 10(2) - 9 = -4 + 20 - 9 = 7 |

| 3 | -3^2 + 10(3) - 9 = -9 + 30 - 9 = 12 |

| 4 | -4^2 + 10(4) - 9 = -16 + 40 - 9 = 15 |

| 5 | -5^2 + 10(5) - 9 = -25 + 50 - 9 = 16 |

| 6 | -6^2 + 10(6) - 9 = -36 + 60 - 9 = 15 |

| 7 | -7^2 + 10(7) - 9 = -49 + 70 - 9 = 12 |

| 8 | -8^2 + 10(8) - 9 = -64 + 80 - 9 = 7 |

| 9 | -9^2 + 10(9) - 9 = -81 + 90 - 9 = 0 |

| 10| -10^2 + 10(10) - 9 = -100 + 100 - 9 = -9 |

The graph is a parabola opening downwards. The vertex is the highest point, so the maximum height is 16 yards. Since the launch is from the one-yard line, the water balloon lands at $x=9$, so it traveled $9-1 = 8$ yards.

3. Final Answer

The water balloon reached a maximum height of 16 yards. It landed 8 yards from where it was launched (at the 9-yard line). The vertex of the parabola represents the maximum height, and the roots (x-intercepts) represent where the balloon lands. Since the launch point is the 1-yard line, we subtract 1 from the root that gives the landing position, which is

9. Therefore, the distance traveled is 8 yards.