Question

a circle representing a pool is graphed with a center at the origin. grant enters the pool at point a and swims over to a friend who is located at point b.
which equation represents grant’s path?
\\(\boldsymbol{y = 2 - 4x}\\)
\\(\boldsymbol{y = 4 - \frac{x}{2}}\\)
\\(\boldsymbol{y = 6 - \frac{x}{4}}\\)
\\(\boldsymbol{y = 8 - 2x}\\)
graph: a circle centered at (0,0) with radius 8 (intercepts at (8,0), (0,8), (-8,0), (0,-8)). line segment from a(8,0) to b(-4,6) on the circle.

Explanation:

Step1: Identify points A and B

From the graph, point A is at \((8, 0)\) and point B is at \((-4, 6)\).

Step2: Calculate the slope (m)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(8,0)\) and \((x_2,y_2)=(-4,6)\). Then \(m=\frac{6 - 0}{-4 - 8}=\frac{6}{-12}=-\frac{1}{2}\).

Step3: Use point - slope form

The point - slope form of a line is \(y - y_1=m(x - x_1)\). Using point A \((8,0)\) and \(m =-\frac{1}{2}\), we have:
\(y-0=-\frac{1}{2}(x - 8)\)
\(y=-\frac{1}{2}x + 4\)
\(y = 4-\frac{x}{2}\)

Answer:

\(y = 4-\frac{x}{2}\) (the option: \(y = 4-\frac{x}{2}\))