Question

1. a line with a positive slope lies in which quadrants?

only in quadrant i
only in quadrants i and ii
quadrants i and iii
quadrants ii and iv

Explanation:

Step1: Recall the properties of slope

The slope of a line \(m=\frac{y_2 - y_1}{x_2 - x_1}\). A positive - slope means that as \(x\) increases, \(y\) increases.

Step2: Analyze the quadrants

• Quadrant I: \(x>0,y>0\).
• Quadrant II: \(x < 0,y>0\).
• Quadrant III: \(x<0,y < 0\).
• Quadrant IV: \(x>0,y < 0\).

For a line \(y = mx + b\) (\(m>0\)):

• If \(b>0\) (y - intercept positive), when \(x = 0,y=b>0\) (point \((0,b)\) in Quadrant I). As \(x\) decreases (becomes negative), \(y=mx + b\) (since \(m>0,x<0\)), if \(|x|\) is large enough, \(y=mx + b\) can be negative. For example, take \(y=x + 1\). When \(x=-2,y=-2 + 1=-1\) (point \((-2,-1)\) in Quadrant III).
• If \(b = 0\) (line passes through the origin), \(y=mx\). When \(x>0,y>0\) (Quadrant I) and when \(x<0,y<0\) (Quadrant III).
• If \(b<0\) (y - intercept negative), when \(x = 0,y=b<0\) (point \((0,b)\) in Quadrant IV). As \(x\) increases, \(y=mx + b\) (since \(m>0\)) will cross into positive \(y\) values. For example, \(y=x-1\). When \(x = 2,y=2-1 = 1\) (point \((2,1)\) in Quadrant I), when \(x=-1,y=-1 - 1=-2\) (point \((-1,-2)\) in Quadrant III)

Answer:

Quadrants I and III