Question

2-81. while watering her outdoor plants, maura noticed that the water coming out of her garden hose followed a parabolic path. thinking that she might be able to model the path of the water with an equation, she quickly took some measurements. the highest point the water reached was 8 feet, and it landed on the plants 10 feet from where she was standing. both the nozzle of the hose and the top of the flowers were 4 feet above the ground. help maura write an equation that describes the path of the water from the hose to the top of her plants. what domain and range make sense for the model?
2-82. draw the graph of $y=2x^{2}+3x+1$.

Explanation:

Step1: Define parabola vertex form

The vertex form of a parabola is $y = a(x-h)^2 + k$, where $(h,k)$ is the vertex. Here, the highest point (vertex) is $(5,8)$ (since it lands 10 feet from the start, the horizontal midpoint is $\frac{10}{2}=5$ feet, height 8 feet). Substitute $h=5, k=8$:
$y = a(x-5)^2 + 8$

Step2: Solve for coefficient $a$

The parabola passes through $(0,4)$ (starting point, 4 ft height at $x=0$). Substitute $x=0, y=4$:
$4 = a(0-5)^2 + 8$
$4 = 25a + 8$
$25a = 4 - 8 = -4$
$a = -\frac{4}{25}$

Step3: Write final equation

Substitute $a=-\frac{4}{25}$ back into the vertex form:
$y = -\frac{4}{25}(x-5)^2 + 8$
Expand to standard form (optional):
$y = -\frac{4}{25}(x^2 -10x +25) +8$
$y = -\frac{4}{25}x^2 + \frac{8}{5}x -4 +8$
$y = -\frac{4}{25}x^2 + \frac{8}{5}x +4$

Step4: Determine domain

The water travels from the hose ($x=0$) to the plants ($x=10$). So domain: $0 \leq x \leq 10$

Step5: Determine range

The minimum height is 4 ft (start/end), maximum is 8 ft (vertex). So range: $4 \leq y \leq 8$

Answer:

Equation: $y = -\frac{4}{25}(x-5)^2 + 8$ (or $y = -\frac{4}{25}x^2 + \frac{8}{5}x + 4$)
Domain: $[0, 10]$ (all real numbers from 0 to 10 feet)
Range: $[4, 8]$ (all real numbers from 4 to 8 feet)