Question

analyzing the graph of a linear function
consider the graph of the linear function ( h(x) = -6 + \frac{2}{3}x ). which quadrant will the graph not go through and why?

• quadrant i, because the slope is negative and the ( y )-intercept is positive
• quadrant ii, because the slope is positive and the ( y )-intercept is negative
• quadrant iii, because the slope is negative and the ( y )-intercept is positive
• quadrant iv, because the slope is positive and the ( y )-intercept is negative

Explanation:

Step1: Identify slope and y-intercept

The linear function is $h(x) = -6 + \frac{2}{3}x$, which can be rewritten as $h(x) = \frac{2}{3}x - 6$. Here, slope $m = \frac{2}{3}$ (positive), y-intercept $b = -6$ (negative).

Step2: Analyze quadrant passage

A line with positive slope rises from left to right. A negative y-intercept means it crosses the y-axis below the origin. Starting from $(0, -6)$ (Quadrant IV) and rising right, it goes through Quadrant I; moving left from $(0, -6)$, it goes through Quadrant III. It never enters Quadrant II.

Answer:

Quadrant II, because the slope is positive and the y-intercept is negative