Problem 4 - College Algebra with Modeling and Visualization

Question

Find the point-slope form of the line passing through the given points. Use the first point as \(\left(x_{1}, y_{1}\right) .\) Plot the points and graph the line by hand. $$ (-1,2),(-2,-3) $$

Step-by-Step Solution

Answer

The point-slope form is \( y - 2 = 5(x + 1) \).

Step 1: Calculate the Slope

The first step is to find the slope of the line that passes through the given points \((-1, 2)\) and \((-2, -3)\). The formula for the slope (\(m\)) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Plugging in the coordinates of the points, we have:\[ m = \frac{-3 - 2}{-2 + 1} = \frac{-5}{-1} = 5 \]So the slope \(m\) is 5.

Step 2: Write the Point-Slope Form Equation

Next, we use the point-slope form of a line equation, which is:\[ y - y_1 = m(x - x_1) \]Here, \(x_1, y_1\) is the first point \((-1, 2)\) and the slope \(m\) is 5. Plug these into the formula:\[ y - 2 = 5(x + 1). \]This equation represents the line in point-slope form.

Step 3: Verify by Plotting the Points

To ensure the equation is correct, plot the two points \((-1, 2)\) and \((-2, -3)\) on a graph. Then, from the point \((-1, 2)\), use the slope \(5\) to rise 5 units and run 1 unit to the right to find another point on the line. Continue checking multiple points if necessary. These plots should fall on a straight line following the equation, confirming the accuracy.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is crucial in mathematics. The slope essentially measures how steep a line is. This is a key concept when determining the relationship between two points on a line. Given two points, you can find the slope using the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points. The slope \(m\) tells us how much \(y\) changes when \(x\) increases by one unit.

• A positive slope means the line ascends from left to right.
• A negative slope indicates the line descends from left to right.
• If the slope is zero, the line is horizontal.
• An undefined slope suggests a vertical line.

In our example, inserting \(x_1 = -1, y_1 = 2, x_2 = -2, y_2 = -3\) into the formula gives us a slope of 5, indicating the line rises sharply from left to right.

Graphing Lines

Graphing lines involves visually representing them on a coordinate plane based on their mathematical equation. One efficient way to graph a line is to use its slope and intercepts. However, when given the point-slope form, you can start from a known point and use the slope to find additional points.
To graph using a line's slope:

• Start at a known point, like \((-1, 2)\).
• From there, use the slope to find subsequent points. If the slope is 5, move up 5 units and to the right 1 unit to plot the next point.

Graphing helps visualize the relationship and confirm its characteristics, like slope direction and line progression across the plane.

Two-Point Formula

The two-point formula is instrumental in creating the equation of a line from two specific points. It involves calculating the slope as the first step, followed by plugging it into the point-slope form of the line equation.
Here's a recap of the important steps:

• Calculate the slope \(m\) using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
• Choose one of the points to serve as \(x_1, y_1\).
• Use the point-slope form: \(y - y_1 = m(x - x_1)\) to create the line's equation.

In the original exercise, the equation derived was \(y - 2 = 5(x + 1)\), illustrating how a line could be defined simply with two points.

Plotting Points

Plotting points accurately on a graph is foundational for visualizing equations and understanding geometric relationships. Each point on a graph is represented by an \(x, y\) pair on the coordinate plane.
Let's break down the process of plotting:

• Identify the \(x\) (horizontal) and \(y\) (vertical) coordinates.
• Start from the origin, \(0,0\), and move horizontally to \(x\).
• Next, move vertically to \(y\).
• Mark the point clearly on the graph.

In the example given, you plot \((-1, 2)\) by moving left 1 unit and up 2 units.
Repeat the process for the other point \((-2, -3)\), moving left 2 units and down 3 units. Accurate plotting keeps the graph true to the line's equation and reflects it correctly in visual representation.