Question

which equation represents the graphed function?
○ $y = -2x + 3$
○ $y = 2x + 3$
○ $y = \frac{1}{2}x + 3$
○ $y = -\frac{1}{2}x + 3$
(graph with points (0, 3) and (1, 1) plotted, and a line passing through them, with x-axis from -5 to 5 and y-axis from -5 to 5.)

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 \((0,3)\), so \(b = 3\).

Step2: Calculate the slope

The slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). We have two points \((0,3)\) and \((1,1)\). Substituting these points into the slope formula: \(m=\frac{1 - 3}{1 - 0}=\frac{-2}{1}=- 2\).

Step3: Identify the equation

Now that we know \(m=-2\) and \(b = 3\), the equation of the line in slope - intercept form is \(y=-2x + 3\). We can also check by substituting \(x = 1\) into the equation \(y=-2x + 3\): \(y=-2(1)+3=-2 + 3 = 1\), which matches the point \((1,1)\) on the graph.

Answer:

\(y=-2x + 3\) (the first option: \(y = - 2x+3\))