Question

graph the line.
$y - 2 = \frac{1}{4}(x + 3)$

Explanation:

Step1: Identify the form of the equation

The given equation \( y - 2=\frac{1}{4}(x + 3) \) is in the point - slope form \( y - y_1=m(x - x_1) \), where \( m=\frac{1}{4} \) (the slope) and the point \( (x_1,y_1)=(- 3,2) \) (since \( y - y_1=y - 2 \) and \( x - x_1=x+3=x-(-3) \)).

Step2: Plot the point

First, locate the point \( (-3,2) \) on the coordinate plane. Move 3 units to the left of the origin along the x - axis (because \( x=-3 \)) and then 2 units up along the y - axis (because \( y = 2 \)) to mark the point \( (-3,2) \).

Step3: Use the slope to find another point

The slope \( m=\frac{1}{4}=\frac{\text{rise}}{\text{run}} \). This means that from the point \( (-3,2) \), we can move 1 unit up (rise) and 4 units to the right (run) to get another point.
Starting from \( (-3,2) \), adding 4 to the x - coordinate (\( - 3+4 = 1 \)) and adding 1 to the y - coordinate (\( 2 + 1=3 \)) gives the point \( (1,3) \). We can also move 1 unit down and 4 units to the left: from \( (-3,2) \), \( x=-3-4=-7 \) and \( y = 2-1 = 1 \), giving the point \( (-7,1) \).

Step4: Draw the line

Draw a straight line passing through the two (or more) points we have found (e.g., \( (-3,2) \) and \( (1,3) \) or \( (-3,2) \) and \( (-7,1) \)). The line should extend in both directions.

Answer:

To graph the line \( y - 2=\frac{1}{4}(x + 3) \):

1. Plot the point \( (-3,2) \).
2. Use the slope \( \frac{1}{4} \) (rise = 1, run = 4) to find a second point (e.g., from \( (-3,2) \), move 1 up and 4 right to get \( (1,3) \)).
3. Draw a straight line through the plotted points.