Question

graph the line with the equation $y = -\frac{1}{2}x + 4$.

Explanation:

Step1: Identify the y-intercept

The equation is in slope - intercept form \(y = mx + b\), where \(b\) is the y - intercept. For \(y=-\frac{1}{2}x + 4\), \(b = 4\). So the line crosses the y - axis at the point \((0,4)\). Mark the point \((0,4)\) on the graph.

Step2: Determine the slope

The slope \(m=-\frac{1}{2}\). The slope is \(\frac{\text{rise}}{\text{run}}\), so from the y - intercept \((0,4)\), we can go down 1 unit (because the numerator of the slope is - 1, which means a decrease) and then 2 units to the right (because the denominator of the slope is 2, which means a positive run). This gives us the next point \((0 + 2,4-1)=(2,3)\). We can also go up 1 unit and 2 units to the left (since a negative slope can also be thought of as \(\frac{1}{-2}\)), so from \((0,4)\), moving left 2 units and up 1 unit gives the point \((0 - 2,4 + 1)=(-2,5)\).

Step3: Draw the line

Connect the points (such as \((0,4)\), \((2,3)\), \((-2,5)\)) with a straight line. The line should extend in both directions.

Answer:

The line is graphed by first plotting the y - intercept \((0,4)\), then using the slope \(m =-\frac{1}{2}\) to find additional points (e.g., \((2,3)\), \((-2,5)\)) and then drawing a straight line through these points.