Question

modeling an equation of a quadratic - quadratic system
in the same coordinate system, a motorboat starts at (2, 3) and travels toward the island along a path that can be modeled with a quadratic function with a vertex at (-1, -1.5).
(x, y) = the boat’s position
vertex form of a quadratic equation:
y = a(x - h)² + k
what equation models the path of the motorboat in the coordinate system?
y = 0.5(x - x + 1)² + 1.5 (with some check and cross marks as shown in the image)

Explanation:

Step1: Identify vertex values

Vertex $(h,k)=(-1,-1.5)$

Step2: Substitute vertex into vertex form

$y=a(x-(-1))^2 + (-1.5)$ simplifies to $y=a(x+1)^2 -1.5$

Step3: Substitute point (2,3) to find $a$

$3=a(2+1)^2 -1.5$
$3=9a -1.5$
$9a=3+1.5=4.5$
$a=\frac{4.5}{9}=0.5$

Step4: Write final equation

Substitute $a=0.5$, $h=-1$, $k=-1.5$ into vertex form.

Answer:

$y=0.5(x+1)^2 -1.5$