Question

graph the equation $y = \frac{1}{2}x + 3$.

Explanation:

Step1: Identify the slope-intercept form

The equation \( y = \frac{1}{2}x + 3 \) is in slope - intercept form \( y=mx + b \), where \( m=\frac{1}{2} \) (slope) and \( b = 3 \) (y - intercept).

Step2: Plot the y - intercept

The y - intercept \( b = 3 \) means the line crosses the y - axis at the point \( (0,3) \). So we mark the point \( (0,3) \) on the coordinate plane.

Step3: Use the slope to find another point

The slope \( m=\frac{1}{2}=\frac{\text{rise}}{\text{run}} \). From the point \( (0,3) \), we can rise (move up) 1 unit and run (move right) 2 units. So we get the point \( (0 + 2,3+1)=(2,4) \)? Wait, no, wait. Wait, the slope is \( \frac{1}{2} \), so from \( (0,3) \), if we move right 2 units (run = 2) and up 1 unit (rise = 1), we get to \( (2,4) \). Alternatively, we can move left 2 units and down 1 unit. Let's check with \( x = 2 \), \( y=\frac{1}{2}(2)+3=1 + 3=4 \), so the point \( (2,4) \) is on the line. The given graph in the problem has a wrong line (it's a negative slope line, but our slope is positive). So to graph the correct line:

• Start at \( (0,3) \) (y - intercept).
• Since the slope is \( \frac{1}{2} \), for every 2 units we move to the right along the x - axis, we move up 1 unit along the y - axis. So from \( (0,3) \), moving right 2 units (to \( x = 2 \)) and up 1 unit (to \( y=4 \)) gives the point \( (2,4) \).
• We can also move left 2 units from \( (0,3) \) (to \( x=- 2 \)) and down 1 unit (to \( y = 2 \)) to get the point \( (-2,2) \).
• Then we draw a straight line through the points \( (0,3) \), \( (2,4) \), \( (-2,2) \) etc.

Answer:

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

1. Plot the y - intercept at \( (0,3) \).
2. Use the slope \( \frac{1}{2} \) to find another point: from \( (0,3) \), move right 2 units and up 1 unit to get \( (2,4) \) (or left 2 units and down 1 unit to get \( (-2,2) \)).
3. Draw a straight line through these points. The line in the given graph is incorrect (it has a negative slope) as the correct line for \( y=\frac{1}{2}x + 3 \) has a positive slope. The correct line should pass through \( (0,3) \), \( (2,4) \), \( (-2,2) \) etc.