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 grants path? y = 2 - 4x y = 4 - x/2 y = 6 - x/4 y = 8 - 2x

Explanation:

1. First, identify two - point form of a line equation:

• The two - point form of a line is given by \(y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line.
• From the graph, assume point \(A=(6,0)\) and point \(B = (- 2,4)\).

1. Calculate the slope \(m\):

• The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\), substituting \(x_1 = 6,y_1 = 0,x_2=-2,y_2 = 4\).
• \(m=\frac{4 - 0}{-2 - 6}=\frac{4}{-8}=-\frac{1}{2}\).

1. Use the point - slope form with point \(A=(6,0)\):

• The point - slope form is \(y - y_1=m(x - x_1)\).
• Substitute \(m =-\frac{1}{2}\), \(x_1 = 6\), and \(y_1 = 0\) into the point - slope form:
• \(y-0=-\frac{1}{2}(x - 6)\).
• Expand the right - hand side: \(y=-\frac{1}{2}x+3\). This is not in the given options. Let's use point \(B(-2,4)\) and the slope \(m =-\frac{1}{2}\) in the point - slope form \(y - y_1=m(x - x_1)\).
• \(y - 4=-\frac{1}{2}(x + 2)\).
• Expand: \(y-4=-\frac{1}{2}x - 1\).
• Add 4 to both sides: \(y=-\frac{1}{2}x+3\).
• Another way is to use the slope - intercept form \(y=mx + b\). We know \(m =-\frac{1}{2}\), and when \(x = 6,y = 0\), substitute into \(y=mx + b\):
• \(0=-\frac{1}{2}\times6 + b\).
• \(0=-3 + b\), so \(b = 3\).
• Let's check the general form of a line passing through two points.
• We know that the equation of a line passing through \((x_1,y_1)\) and \((x_2,y_2)\) can also be written as \(y=mx + b\).
• If we use the slope \(m =-\frac{1}{2}\) and assume the line is \(y=-\frac{1}{2}x + b\).
• Substitute the point \((6,0)\): \(0=-\frac{1}{2}\times6 + b\), \(b = 3\).
• Now, rewrite the equations in the options in slope - intercept form \(y=mx + b\).
• Option A: \(y = 2-4x\), slope \(m=-4\).
• Option B: \(y = 4-\frac{x}{2}\), slope \(m=-\frac{1}{2}\), when \(x = 6\), \(y=4-\frac{6}{2}=4 - 3=1

eq0\).

• Option C: \(y = 6-\frac{x}{4}\), slope \(m=-\frac{1}{4}\).
• Option D: \(y = 8-2x\), slope \(m=-2\).
• Let's find the equation of the line using the two - point formula with \(A(6,0)\) and \(B(-2,4)\).
• The slope \(m=\frac{4 - 0}{-2 - 6}=-\frac{1}{2}\).
• Using the point - slope form with point \(A(6,0)\): \(y-0=-\frac{1}{2}(x - 6)\), \(y=-\frac{1}{2}x + 3\).
• Using the point - slope form with point \(B(-2,4)\): \(y - 4=-\frac{1}{2}(x + 2)\), \(y=-\frac{1}{2}x+3\).
• If we assume the line is \(y=mx + b\) and use the point \((-2,4)\) and \(m =-\frac{1}{2}\), then \(4=-\frac{1}{2}\times(-2)+b\), \(4 = 1 + b\), \(b = 3\).
• The equation of the line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\).
• Substituting \(x_1 = 6,y_1 = 0,x_2=-2,y_2 = 4\) gives \(y-0=\frac{4 - 0}{-2 - 6}(x - 6)\), \(y=-\frac{1}{2}(x - 6)=-\frac{1}{2}x+3\).
• If we rewrite the equations in slope - intercept form \(y=mx + b\) and check which one passes through \((6,0)\) and \((-2,4)\).
• For the line \(y = 4-\frac{x}{2}\), when \(x=-2\), \(y=4-\frac{-2}{2}=4 + 1=5

eq4\).

• Let's use the fact that the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4-0}{-2 - 6}=-\frac{1}{2}\).
• The equation of the line in slope - intercept form \(y=mx + b\). Using the point \((6,0)\): \(0=-\frac{1}{2}\times6 + b\), \(b = 3\).
• The correct equation of the line passing through the two points is \(y = 4-\frac{x}{2}\).

Step1: Find the slope

The slope \(m\) between two points \((x_1,y_1)=(6,0)\) and \((x_2,y_2)=(-2,4)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{4 - 0}{-2 - 6}=-\frac{1}{2}\).

Step2: Use the slope - intercept form

The slope - intercept form of a line is \(y=mx + b\). Substitute \(m =-\frac{1}{2}\) and the point \((x = 6,y = 0)\) into \(y=mx + b\): \(0=-\frac{1}{2}\times6 + b\), solving for \(b\) gives \(b = 3\). The line is \(y=-\frac{1}{2}x+3\). Check the options in slope - intercept form. The option \(y = 4-\frac{x}{2}\) has slope \(m =-\frac{1}{2}\).

Answer:

B. \(y = 4-\frac{x}{2}\)