Problem 36 - Precalculus with Limits

Question

In Exercises 29-40, plot the points and find the slope of the line passing through the pair of points. \((0, -10)\), \((-4, 0)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (0, -10) and (-4, 0) is -2.5.

Step 1: Identify the Given Points

The given points are (0, -10) and (-4, 0). These are the coordinates (x1, y1) and (x2, y2) respectively.

Step 2: Use the Slope Formula

Substitute the coordinates of the points into the slope formula, \(m = (y2-y1) / (x2-x1)\). This gives \(m = (0 - -10) / (-4 - 0)\).

Step 3: Simplify the Expression

Solve the equation \(m = 10 / -4\). This simplifies to \(m = -2.5\).

Key Concepts

Plotting Points

Understanding how to plot points is essential for visualizing mathematical concepts like lines, functions, and shapes on a coordinate plane. To start plotting a point, you need a pair of coordinates, typically written as \( (x, y) \), where \( x \) is the value on the horizontal axis and \( y \) is the value on the vertical axis.

To plot the points from our example, \( (0, -10) \) and \( (-4, 0) \) on a coordinate plane, first locate the origin (the point where \( x \) and \( y \) both equal zero). For the first point \( (0, -10) \) start at the origin, move zero units along the x-axis (since the x-coordinate is 0) and then move down 10 units along the y-axis because the y-coordinate is -10. For the second point \( (-4, 0) \) move 4 units to the left of the origin (since the x-coordinate is -4) and no units up or down the y-axis (since the y-value is 0).

Plotting these points accurately is crucial as they serve as reference points for drawing lines, shaping parabolas, or solving geometric problems.

Slope Formula

The slope of a line is a measure of its steepness and direction. It is a key concept when dealing with linear equations. The slope formula is used to calculate this value and is represented by \( m = (y2 - y1) / (x2 - x1) \) where \( m \) is the slope, \( (x1, y1) \) and \( (x2, y2) \) are the coordinates of two points on the line.

Using our given points, \( (0, -10) \) and \( (-4, 0) \) as \( (x1, y1) \) and \( (x2, y2) \) respectively, the slope can be calculated as follows: plug in the values into the formula, which gives us \( m = (0 - (-10)) / (-4 - 0) \) or \( m = 10 / -4 \) after simplification. This calculation results in \( m = -2.5 \) indicating that for every 4 units we move horizontally to the left, the line will drop down by 2.5 units.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by two number lines intersecting at a right angle. The horizontal line is called the x-axis, and the vertical line is called the y-axis. The point at which they intersect is known as the origin \( (0, 0) \).

The entire plane is divided into four quadrants, each representing a unique combination of positive and negative values of \( x \) and \( y \) coordinates. In the context of our example, the point \( (0, -10) \) lies on the negative side of the y-axis (also referred to as the vertical axis), which is in the fourth quadrant, while the point \( (-4, 0) \) lies on the negative side of the x-axis (the horizontal axis), which is in the second quadrant. Knowing how to navigate the coordinate plane is crucial to graphing equations, solving inequalities, and understanding the geometric representation of functions.