To determine which quadrant the graph of the linear function [tex]\( h(x) = -6 + \frac{2}{3} x \)[/tex] will not go through, we follow these steps:
1. Identify the slope and y-intercept:
The given linear function is in the form [tex]\( h(x) = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
[tex]\[ m = \frac{2}{3}, \][/tex]
[tex]\[ b = -6. \][/tex]
2. Understand the y-intercept:
The y-intercept [tex]\( b \)[/tex] is the point where the graph crosses the y-axis. Here, it is [tex]\( -6 \)[/tex], which is below the origin on the y-axis.
3. Interpret the slope:
The slope [tex]\( \frac{2}{3} \)[/tex] is positive, meaning the line rises as it moves from left to right.
4. Determine which quadrants the line will pass through based on the slope and y-intercept:
- Since the y-intercept is negative ([tex]\( -6 \)[/tex]), the graph starts below the origin on the y-axis.
- As the slope is positive ([tex]\( \frac{2}{3} \)[/tex]), the line will rise as it moves rightward.
Therefore:
- Starting in the negative y-region (below the x-axis), the graph will initially be in Quadrant III when moving from left to right.
- Because the slope is positive, the line will move up and to the right but still remain below the x-axis, entering Quadrant IV.
5. Which quadrants does the graph never touch?:
- The graph rises but starts below the x-axis and crosses the x-axis to enter Quadrant IV directly.
- It never crosses into the quadrants where [tex]\( y \)[/tex] is positive (Quadrants I and II) since the y-intercept is negative and the slope is insufficient to bring [tex]\( y \)[/tex] positive.
With this understanding, the correct answer to which quadrant the graph will not pass through given its positive slope and negative y-intercept is:
Quadrant I, because the slope is positive and the y-intercept is negative.
