Problem 1 - Precalculus: Mathematics for Calculus

Question

We find the "steepness," or slope, of a line passing through two points by dividing the difference in the __________ -coordinates of these points by the difference in the _________ -coordinates. So the line passing through the points \((0,1)\) and \((2,5)\) has slope __________.

Step-by-Step Solution

Answer

The slope of the line is 2.

Step 1: Identify the Two Points

We are given two points: \((0,1)\) and \((2,5)\). Each of these points has an \(x\)-coordinate and a \(y\)-coordinate.

Step 2: Formula for Slope

The formula to find the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 3: Substitute Coordinates

Substitute the coordinates into the slope formula: \(m = \frac{5 - 1}{2 - 0}\).

Step 4: Calculate the Differences

Calculate the difference in the \(y\)-coordinates: \(5 - 1 = 4\). Next, calculate the difference in the \(x\)-coordinates: \(2 - 0 = 2\).

Step 5: Compute the Slope

Divide the difference in the \(y\)-coordinates by the difference in the \(x\)-coordinates: \(\frac{4}{2} = 2\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric principles. It involves representing geometric shapes in a system of coordinates, commonly a coordinate plane, where each point is defined by an ordered pair \((x, y)\). By plotting these coordinates, we can visualize and solve geometric problems. This fusion of algebra and geometry allows for precise calculations and deeper insights into spatial relationships, such as finding the distance between points or the slope of a line.
Coordinate geometry is practical because it helps translate geometric problems into algebraic ones, making it easier to use algebraic tools to solve them.

Points on a Plane

In coordinate geometry, points on a plane are crucial. Each point is represented by two numbers written as an ordered pair, \((x, y)\). The first number, \(x\), is the x-coordinate, while the second number, \(y\), is the y-coordinate.
These numbers determine the position of a point in a two-dimensional space defined by the x-axis (horizontal) and y-axis (vertical). A point's location is found by moving along the x-axis to the x-coordinate and then vertically to the y-coordinate.

• Origin: The point \((0, 0)\) where the x-axis and y-axis meet.
• Quadrants: The coordinate plane is divided into four quadrants by the x-axis and y-axis.

Understanding the exact placement of points on a plane is fundamental to solving problems involving lines and slopes.

Calculating Slope

The slope of a line is a measure of its steepness and direction. It's calculated using two points on the line. The formula for the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
This process involves:

• Identifying two points on the line.
• Finding the difference in the y-coordinates, \(y_2 - y_1\).
• Finding the difference in the x-coordinates, \(x_2 - x_1\).
• Dividing the difference in y by the difference in x to find the slope.

The slope tells us how much the line rises or falls for each unit it moves horizontally. A positive slope means the line rises as it moves to the right, while a negative slope means it falls.

x-coordinate

The x-coordinate of a point is its horizontal value on the coordinate plane. It's the first number in the ordered pair \((x, y)\). This coordinate tells us how far along the x-axis the point is.
When determining the position of a point, the x-coordinate is used to "walk" left or right from the origin, depending on whether it's positive or negative. For example, in the point \((2, 5)\), the x-coordinate is 2, indicating the point is 2 units to the right of the y-axis.

• If the x-coordinate is zero, the point lies directly on the y-axis.
• Positive x-coordinates are to the right of the origin.
• Negative x-coordinates are to the left of the origin.

The significance of the x-coordinate becomes clear when calculating slope, as it's part of the formula's denominator, \(x_2 - x_1\).

y-coordinate

The y-coordinate signifies a point's vertical position on a coordinate plane and is the second number in the ordered pair \((x, y)\). This coordinate informs us how far up or down a point is from the x-axis.
In navigation from the origin, you use the y-coordinate to move up if it's positive, and down if it's negative. For instance, in the point \((0, 1)\), the y-coordinate is 1, meaning the point is 1 unit above the x-axis.

• If the y-coordinate is zero, the point lies directly on the x-axis.
• Positive y-coordinates lie above the x-axis.
• Negative y-coordinates lie below the x-axis.

In slope calculation, the y-coordinate helps to determine the line's rise relative to a change along the horizontal line.