Question

which equation represents the graphed function?
\\( y = 4x - 2 \\)
\\( y = -4x - 2 \\)
\\( y = \frac{1}{4}x - 2 \\)
\\( y = -\frac{1}{4}x - 2 \\)
(graph shows a line passing through (0, -2) and (4, -1))

Explanation:

Step1: Recall slope-intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. From the graph, we can see that the line crosses the y - axis at \((0,-2)\), so \(b=-2\).

Step2: Calculate the slope \(m\)

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). We have two points on the line: \((0,-2)\) and \((4,-1)\). Let \((x_1,y_1)=(0,-2)\) and \((x_2,y_2)=(4,-1)\). Then \(m=\frac{-1-(-2)}{4 - 0}=\frac{-1 + 2}{4}=\frac{1}{4}\).

Step3: Identify the equation

Since \(m = \frac{1}{4}\) and \(b=-2\), the equation of the line in slope - intercept form is \(y=\frac{1}{4}x-2\). We can also check the sign of the slope. The line is increasing (from left to right, it goes up), so the slope should be positive. This eliminates the options with \(m=-4\) and \(m =-\frac{1}{4}\). And since \(m=\frac{1}{4}\) (not 4), we can confirm that the equation is \(y=\frac{1}{4}x - 2\).

Answer:

\(y=\frac{1}{4}x - 2\) (the third option: \(y=\frac{1}{4}x-2\))