Question

the trajectory of the water from fountain a is represented by a function in standard form while the trajectory of the water from fountain b is represented by a table of values. compare the vertex of each function. which trajectory reaches a greater height in feet?
fountain a
$f(x) = -x^2 + 10x - 18$
fountain b

x	-7	-6	-5	-4	-3
y	1	7	9	7	1

fountain a has a vertex of \\(\square\\) while fountain b has a vertex of \\(\square\\), so the water from fountain \\(\blacktriangledown\\) reaches a greater height.
(type ordered pairs.)

Explanation:

Step1: Find vertex of Fountain A

The function for Fountain A is $f(x) = -x^2 + 10x - 18$. For a quadratic function $ax^2 + bx + c$, the x - coordinate of the vertex is $x = -\frac{b}{2a}$. Here, $a=-1$, $b = 10$. So $x=-\frac{10}{2\times(-1)} = 5$. Substitute $x = 5$ into the function: $f(5)=-(5)^2+10\times5 - 18=-25 + 50-18 = 7$. So the vertex of Fountain A is $(5,7)$.

Step2: Find vertex of Fountain B

For Fountain B, we have the table:

x	-7	-6	-5	-4	-3

The parabola is symmetric. The maximum value (vertex) occurs at the middle x - value. The x - values are symmetric around $x=-5$ (since the values of y are symmetric: 1,7,9,7,1). When $x = - 5$, $y = 9$. So the vertex of Fountain B is $(-5,9)$.

Step3: Compare the y - coordinates of the vertices

The y - coordinate of the vertex of Fountain A is 7, and the y - coordinate of the vertex of Fountain B is 9. Since $9>7$, the water from Fountain B reaches a greater height.

Answer:

Fountain A has a vertex of $\boldsymbol{(5,7)}$ while Fountain B has a vertex of $\boldsymbol{(-5,9)}$, so the water from Fountain $\boldsymbol{B}$ reaches a greater height.