Question

a) $m = \frac{2}{-4} = -\frac{1}{2}$
b) $b = -2$
c) $y = -\frac{1}{2}x - 2$

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((-4, 0)\) and \((0, -2)\).

Step2: Calculate the slope (\(m\))

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(-4,0)\) and \((x_2,y_2)=(0, -2)\). Then \(m=\frac{-2 - 0}{0 - (-4)}=\frac{-2}{4}=-\frac{1}{2}\), which matches option a (since \(\frac{2}{-4}=-\frac{1}{2}\)).

Step3: Determine the y - intercept (\(b\))

The y - intercept is the value of \(y\) when \(x = 0\). From the point \((0,-2)\), we see that \(b=-2\)? Wait, no. Wait, the point \((0,-2)\) is on the y - axis, so the y - intercept \(b\) is \(-2\)? Wait, no, wait the line passes through \((0, -1)\)? Wait, maybe I misread the graph. Wait, looking again, the y - intercept: the line crosses the y - axis at \((0, -1)\)? Wait, no, the original graph: let's re - examine. Wait, the first point is \((-4,0)\) and the second point is \((0, - 1)\)? Wait, maybe the user's graph has a typo, but according to the options, let's check option c: \(y =-\frac{1}{2}x-2\). Wait, if we use the slope \(m =-\frac{1}{2}\) and the y - intercept \(b=-1\), but the options have \(b = - 2\) and \(y=-\frac{1}{2}x - 2\). Wait, maybe the two points are \((-4,0)\) and \((0,-2)\). Then the slope is \(\frac{-2 - 0}{0+4}=\frac{-2}{4}=-\frac{1}{2}\), and the y - intercept \(b=-2\) (since when \(x = 0\), \(y=-2\)). Then the equation of the line is \(y=mx + b=-\frac{1}{2}x-2\), which is option c.

Wait, let's check each option:

- Option a: \(m=\frac{2}{-4}=-\frac{1}{2}\), which is correct because the change in \(y\) from \((-4,0)\) to \((0,-2)\) is \(-2\) (or \(2\) in the numerator if we reverse the points) and change in \(x\) is \(4\) (or \(-4\) if we reverse the points), so \(\frac{2}{-4}=-\frac{1}{2}\), so option a is correct.

- Option b: The y - intercept \(b\) is the value of \(y\) when \(x = 0\). If the line crosses the y - axis at \((0,-1)\), then \(b=-1\), but if we take the points as \((-4,0)\) and \((0,-2)\), then \(b=-2\). Wait, maybe the graph is drawn with a grid where each square is 1 unit. Let's assume the two points are \((-4,0)\) and \((0,-2)\). Then \(b=-2\) (option b) is correct? Wait, no, the equation of the line with \(m =-\frac{1}{2}\) and \(b=-2\) is \(y=-\frac{1}{2}x-2\) (option c).

Wait, let's re - evaluate:

1. For slope (\(m\)):
Using points \((x_1,y_1)=(-4,0)\) and \((x_2,y_2)=(0,-2)\)
\(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2 - 0}{0-(-4)}=\frac{-2}{4}=-\frac{1}{2}\)
\(\frac{2}{-4}=-\frac{1}{2}\), so option a is correct.

2. For y - intercept (\(b\)):
The y - intercept is the \(y\) - value when \(x = 0\). From the point \((0,-2)\), \(b=-2\), so option b is correct? Wait, no, if the line is \(y =-\frac{1}{2}x-2\), then when \(x = 0\), \(y=-2\), so \(b=-2\) (option b is correct) and the equation \(y =-\frac{1}{2}x-2\) (option c) is correct. And the slope calculation in option a is correct.

But let's check the options:

- Option a: \(m=\frac{2}{-4}=-\frac{1}{2}\), correct.

- Option b: \(b=-2\), correct (since the y - intercept is at \((0,-2)\)).

- Option c: \(y =-\frac{1}{2}x-2\), correct (using \(y=mx + b\) with \(m =-\frac{1}{2}\) and \(b=-2\)).

But maybe the question is to check which statement is correct. Let's assume the question is to identify the correct statements.

But if we have to choose one, let's see:

For option a: The slope formula can be calculated as \(\frac{y_2 - y_1}{x_2 - x_1}\). If we take the two points as \((-4,0)\) and \((0,-2)\), then the change in \(y\) is \(-2-0=-2\) and change in \(x\) is \(0 - (-4)=4\). But if we reverse the points, change in \(y\) is \(0-(-2)=2\) and change in \(x\) is \(-4 - 0=-4\), so \(m=\frac{2}{-4}=-\frac{1}{2}\), so option a is correct.

For option b: The y - intercept is the value of \(y\) when \(x = 0\). From the graph, if the line crosses the y - axis at \((0,-1)\), then \(b=-1\), but if it crosses at \((0,-2)\), then \(b=-2\). Given the option \(b = - 2\), and the equation in option c is \(y=-\frac{1}{2}x-2\), which would have a y - intercept of \(-2\).

For option c: Using \(m =-\frac{1}{2}\) and \(b=-2\), the equation \(y=mx + b\) gives \(y=-\frac{1}{2}x-2\), which is correct.

Answer:

Assuming we are to check the correct statements:

a) is correct (since \(\frac{2}{-4}=-\frac{1}{2}\)), b) is correct (if the y - intercept is \(-2\)), c) is correct (since \(y =-\frac{1}{2}x-2\) with \(m =-\frac{1}{2}\) and \(b=-2\)).

But if we have to choose one, let's say we are checking the equation of the line. The equation of a line in slope - intercept form is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We found \(m =-\frac{1}{2}\) and \(b=-2\), so \(y =-\frac{1}{2}x-2\) (option c) is correct. Also, the slope calculation in option a is correct (\(m=\frac{2}{-4}=-\frac{1}{2}\)) and the y - intercept in option b: if the line crosses the y - axis at \((0,-2)\), then \(b=-2\) is correct.