To find the vertical intercept \( b \) in the linear equation \( y = mx + b \), we first need to determine the slope \( m \) using the two given points on the graph: \( (1, 5) \) and \( (3, 8) \).
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \( (1, 5) \) and \( (3, 8) \):
- \( x_1 = 1 \), \( y_1 = 5 \)
- \( x_2 = 3 \), \( y_2 = 8 \)
Now let's calculate the slope \( m \):
\[ m = \frac{8 - 5}{3 - 1} = \frac{3}{2} = 1.5 \]
Now that we have the slope, we can use one of the points to solve for \( b \). Let’s use the point \( (1, 5) \).
The equation of a line in slope-intercept form is:
\[ y = mx + b \]
Substituting in the values we know:
\[ 5 = 1.5 \cdot 1 + b \]
This simplifies to:
\[ 5 = 1.5 + b \]
Now we solve for \( b \):
\[ b = 5 - 1.5 \]
\[ b = 3.5 \]
Thus the value of \( b \) (the vertical intercept) is:
\[ \boxed{3.5} \]