Question

which equations and/or functions represent the graphed line? select three options. graph of a line on a coordinate plane with x-axis from -6 to 6 and y-axis from -6 to 6. the line passes through points (-4, 0), (-2, 1), (0, 2), (2, 3), etc. options: \\( f(x) = \frac{1}{5}x - 4 \\), \\( f(x) = \frac{1}{2}x + 2 \\)

Explanation:

Step1: Find the slope

From the graph, two points are \((-4, 0)\) and \((0, 2)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{0 - (-4)}=\frac{2}{4}=\frac{1}{2}\).

Step2: Find the y - intercept

The line crosses the y - axis at \((0, 2)\), so the y - intercept \(b = 2\).

Step3: Write the equation

The slope - intercept form of a line is \(y=mx + b\), so the equation is \(y=\frac{1}{2}x + 2\) or \(f(x)=\frac{1}{2}x + 2\). Now check the first function \(f(x)=\frac{1}{5}x-4\): the slope is \(\frac{1}{5}\) and y - intercept is \(- 4\), which does not match the graph. The second function \(f(x)=\frac{1}{2}x + 2\) has slope \(\frac{1}{2}\) and y - intercept \(2\), which matches the graph. (Assuming there are other options not fully shown, but based on the given two, the correct one among them is \(f(x)=\frac{1}{2}x + 2\))

Answer:

\(f(x)=\frac{1}{2}x + 2\)