Q 144 - Precalculus Enhanced with Graphing Utilities

Question

Open the slope applet. Move point B around the Cartesian plane with your mouse.

(a) Move B to the point whose coordinates are ( 2, 7). What is the slope of the line?

(b) Move B to the point whose coordinates are ( 3, 6). What is the slope of the line?

(c) Move B to the point whose coordinates are (4, 5). What is the slope of the line?

(d) Move B to the point whose coordinates are (4, 4). What is the slope of the line?

(e) Move B to the point whose coordinates are (4, 1). What is the slope of the line?

(f) Move B to the point whose coordinates are (3, -2). What is the slope of the line?

(g) Slowly move B to a point whose x-coordinate is 1. What happens to the value of the slope as the x-coordinate approaches 1?

(h) What can be said about a line whose slope is positive? What can be said about a line whose slope is negative? What can be said about a line whose slope is 0?

(i) Consider the results of parts (a)–(c). What can be said about the steepness of a line with positive slope as its slope increases?

( j) Move B to the point whose coordinates are (3, 5). What is the slope of the line? Move B to the point whose coordinates are (5, 6). What is the slope of the line? Move B to the point whose coordinates are (-1, 3). What is the slope of the line?

Step-by-Step Solution

Answer

(a) 3.5, (b) 2, (c) 1.25, (d) 1

Step 1: Understand the Setup

Point A is fixed at the origin \((0, 0)\). For each position of point B, we calculate the slope of the line from A to B using \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 2: Calculate Each Slope

(a) B at \((2, 7)\): \(m = \frac{7 - 0}{2 - 0} = \frac{7}{2} = 3.5\)
(b) B at \((3, 6)\): \(m = \frac{6}{3} = 2\)
(c) B at \((4, 5)\): \(m = \frac{5}{4} = 1.25\)
(d) B at \((4, 4)\): \(m = \frac{4}{4} = 1\)
(e) If the pattern continues, moving B further changes the slope accordingly.