Problem 62 - Calculus Volume 1

Question

For the following exercises, for each pair of points, a. find the slope of the line passing through the points and b. indicate whether the line is increasing, decreasing, horizontal, or vertical. $$ (6,4) \text { and }(4,-3) $$

Step-by-Step Solution

Answer

Slope = 7/2; Line is increasing.

Step 1: Identify the coordinates of the points

We start by identifying the coordinates of the given points: Point A is (6, 4) and Point B is (4, -3). Here, \( x_1=6, y_1=4, x_2=4, \text{ and } y_2=-3 \).

Step 2: Apply the slope formula

The formula to find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2-y_1}{x_2-x_1} \] Substituting the values, we get: \[ m = \frac{-3-4}{4-6} \]

Step 3: Calculate the slope value

Calculate the value by simplifying the expression: \[ m = \frac{-7}{-2} = \frac{7}{2} \] So, the slope \(m\) is \(\frac{7}{2}\).

Step 4: Determine the line orientation

Because the slope \(m = \frac{7}{2} > 0\) is positive, the line is increasing. This means as one moves from left to right, the line rises.

Key Concepts

Line Orientation

Line orientation refers to the direction in which a line travels on a coordinate plane. Understanding orientation is key to graphing and interpreting linear equations.

• An increasing line rises from left to right. This line has a positive slope, indicating the relationship between the variables is directly proportional.
• A decreasing line falls from left to right, with a negative slope, suggesting an inverse relationship between the variables.
• A horizontal line is flat along the plane, representing zero slope, meaning there is no change in the vertical direction as you move horizontally.
• A vertical line goes straight up and down. Such lines have an undefined slope because division by zero occurs in the slope formula.

Recognizing these orientations helps in predicting how changes in one variable affect the other.

Slope Formula

The slope formula is essential for finding the steepness of a line through two points. It is given by:
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This formula calculates the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \).

• Rise: The difference in the y-coordinates, \( y_2 - y_1 \).
• Run: The difference in the x-coordinates, \( x_2 - x_1 \).

The slope tells us how much Y changes for a one-unit change in X. In our example, we computed a slope of \( \frac{7}{2} \), meaning for every 2 units you go right, the line goes up 7 units.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses the coordinate plane to explore geometric concepts. It bridges algebra and geometry through coordinates.

• The coordinate plane consists of two axes: the x-axis (horizontal) and the y-axis (vertical).
• Every point on the plane is identified by an ordered pair \( (x, y) \).
• This system enables calculating distances, midpoints, and slopes, connecting algebraic equations with geometric shapes.

In our exercise, we used coordinate geometry to find the slope of a line segment between two points. By placing points \( (6, 4) \) and \( (4, -3) \) on the plane, the geometric relationship revealed its algebraic equivalent: the slope. Using these methods can simplify visualising mathematical concepts as graphical representations of equations.