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) = -x² + 2x + 7
fountain b

x	y
-5	5
-4	6
-3	5
-2	2

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

Explanation:

Step1: Find vertex of Fountain A

For quadratic $f(x)=ax^2+bx+c$, vertex $x$-coordinate is $x=-\frac{b}{2a}$.
Here $a=-1, b=2$, so $x=-\frac{2}{2(-1)}=1$.
Substitute $x=1$ into $f(x)$:
$f(1)=-(1)^2+2(1)+7=-1+2+7=8$.
Vertex of Fountain A is $(1,8)$.

Step2: Identify vertex of Fountain B

The table for Fountain B is symmetric around $x=-4$ (values of $y$ mirror on either side of $x=-4$). The maximum $y$-value is 6 at $x=-4$.
Vertex of Fountain B is $(-4,6)$.

Step3: Compare vertex y-values

$8 > 6$, so Fountain A's water is higher.

Answer:

Fountain A has a vertex of $(1, 8)$ while Fountain B has a vertex of $(-4, 6)$, so the water from Fountain A reaches a greater height.