Problem 1 - 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. $$ (1,2),(3,-2) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Points

We are given two points: \((1, 2)\) and \((3, -2)\). We'll use \((1, 2)\) as \((x_1, y_1)\).

Step 2: Calculate the Slope

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_2 = 3\), \(y_2 = -2\), \(x_1 = 1\), and \(y_1 = 2\). Substitute these into the formula: \[ m = \frac{-2 - 2}{3 - 1} = \frac{-4}{2} = -2 \]. The slope \(m\) is \(-2\).

Step 3: Write the Point-Slope Form Equation

The point-slope form of a line is given by \(y - y_1 = m(x - x_1)\). Substitute \(m = -2\), \(x_1 = 1\), and \(y_1 = 2\) into the equation: \[ y - 2 = -2(x - 1) \].

Step 4: Graph the Line

Plot the points \((1, 2)\) and \((3, -2)\) on a graph. Using the slope, draw the line that passes through both points. The line should fall with a slope of \(-2\), meaning it goes down 2 units for every 1 unit it moves to the right.

Key Concepts

Slope Calculation

When trying to understand the concept of a slope, think of it as describing how steep a line is. It measures the vertical change per unit of horizontal change between two points on a line. To calculate the slope, we use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, the subscript numbers (1 and 2) represent the coordinates of two different points, often noted as \((x_1, y_1)\) and \((x_2, y_2)\). Begin by substituting the given points into the formula to find the slope, as demonstrated in the problem:

• Using points \((1, 2)\) and \((3, -2)\), the slope is calculated as \( m = \frac{-2 - 2}{3 - 1} = -2 \).

The result \(-2\) indicates that for every unit you move right along the x-axis, the line slopes downward by 2 units.

Graphing Lines

Graphing lines involves plotting points on a coordinate system and connecting them to visualize the line. Start with identifying your given points, like \((1, 2)\) and \((3, -2)\) from the exercise, and accurately plot them on the graph grid. The x-values dictate the position horizontally, while the y-values decide the vertical positioning.

• Place the point \((1, 2)\) by moving 1 unit right and 2 units up from the origin.
• Next, locate \((3, -2)\) by shifting 3 units to the right and 2 units down.

Once the points are plotted, draw a straight line through them.
The slope of \(-2\) tells you the line falls 2 units for each 1 unit it goes right, thus helping in predicting its direction if it extends further on the graph. This visual representation aids immensely in understanding the slope's impact on the line.

Equation of a Line

The equation of a line in point-slope form is particularly handy when you have a point and a slope. This form is expressed as:

• \( y - y_1 = m(x - x_1) \)

It incorporates a known point, \((x_1, y_1)\), and the slope \(m\), offering a straightforward way to formulate the line's equation. Using our problem's point and slope \((1, 2)\) and \(-2\) respectively, we substitute into the formula:

• \( y - 2 = -2(x - 1) \)

This expression is a concise depiction of the line passing through the given point with the calculated slope.
Point-slope form is extremely useful in further calculations, conversions to other forms like slope-intercept or standard form, and is often the first step in deeper analytical geometry explorations.