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

Step-by-Step Solution

Answer

The point-slope form is \(y + 1 = \frac{3}{4}(x + 3)\).

Step 1: Identify Given Points

The given points are \((-3, -1)\) and \((1, 2)\). We will use \((-3, -1)\) as \((x_1, y_1)\).

Step 2: Find the Slope of the Line

To find the slope \(m\), use the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substitute the given points: \[m = \frac{2 - (-1)}{1 - (-3)} = \frac{3}{4}\]

Step 3: Write Equation in Point-Slope Form

The point-slope form equation is given by: \[y - y_1 = m(x - x_1)\]Substitute \(m = \frac{3}{4}\), \(x_1 = -3\), and \(y_1 = -1\) into the equation:\[y - (-1) = \frac{3}{4}(x - (-3))\] \[y + 1 = \frac{3}{4}(x + 3)\]

Step 4: Plot the Points and Graph the Line

On graph paper, plot the points \((-3, -1)\) and \((1, 2)\). Draw a straight line passing through these points. Use the point-slope equation \(y + 1 = \frac{3}{4}(x + 3)\) to guide the slope and direction of the line.

Key Concepts

Understanding Slope Calculation

Slope is a measure of how steep a line is. It represents the rate at which the `y` value changes with 'x'. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. The formula to calculate the slope `m` of a line using two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

In the example given, the points are \( ( -3, -1) \) and \( (1, 2) \). By substituting these into the slope formula, we find:

• \( m = \frac{2 - (-1)}{1 - (-3)} = \frac{3}{4} \)

This tells us that for every 4 units the line moves horizontally, it moves 3 units vertically. Slope calculation is crucial for understanding the behavior of linear graphs.

Graphing Lines Using Point-Slope Form

Graphing a line using the point-slope form is a practical way to visualize the equation The point-slope form of a line's equation is:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) is a known point on the line, and `m` is the slope.In the exercise, we substitute \( (x_1, y_1) = (-3, -1) \) and the slope \( m = \frac{3}{4} \):

• \( y + 1 = \frac{3}{4}(x + 3) \)

To graph the line, you can use this equation to calculate various points by substituting different `x` values.Then, plot these points on a coordinate plane, and connect them with a straight line.The slope will guide how steep your line is, and this visualization helps understand relationships in real-life situations.

Exploring Coordinate Geometry Concepts

Coordinate geometry, or analytic geometry, is a branch of mathematics dealing with plotting points, lines, and figures on the coordinate plane. It helps in visualizing relationships between different algebraic equations. The coordinate plane consists of an

• x-axis (horizontal line)
• y-axis (vertical line)

Using these axes, any point can be represented using a pair of coordinates \( (x, y) \).In solving the given exercise, we made use of these principles to:

• Determine the slope from a pair of points
• Utilize the point-slope form for writing equations
• Plot points and graph lines effectively

Coordinate geometry is a powerful tool as it allows us to transition smoothly between algebraic equations and geometric representations, aiding in both theoretical and practical applications.