Problem 73 - College Algebra

Question

For the following exercises, sketch a line with the given features. A \(y\) -intercept of (0,3) and slope \(\frac{2}{5}\)

Step-by-Step Solution

Answer

Plot (0,3), move right 5, up 2, through (5,5), draw the line.

Step 1: Identify Key Features

The line has a specified y-intercept at the point (0,3) and a slope of \( \frac{2}{5} \). Note these values are essential for plotting the line.

Step 2: Plot the Y-Intercept

Firstly, locate the y-intercept (0,3) on the Cartesian plane and mark this point. This is where your line will intersect the y-axis.

Step 3: Understand the Slope

The slope \( \frac{2}{5} \) indicates that for every 5 units moved horizontally to the right, the line moves up 2 units. This ratio will help to plot additional points.

Step 4: Plot Additional Points Using the Slope

Starting at the y-intercept (0,3), move right along the x-axis 5 units to x = 5, then move up 2 units to the point (5,5). Mark this point.

Step 5: Sketch the Line

Draw a straight line through the points (0,3) and (5,5). Extend the line across the graph, ensuring it reflects the determined slope and y-intercept.

Key Concepts

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis on a Cartesian plane. It is an essential component in graphing linear equations because it gives us a starting point. When the equation of a line is in the form \( y = mx + b \), the y-intercept is \( b \). In our exercise, the line has a y-intercept at the point (0,3). This tells us where the line meets the vertical y-axis, providing a reference point for plotting the rest of the line. By identifying this point, we begin sketching our line accurately.

Slope

The slope of a line measures its steepness and direction. It is often denoted by \( m \) in the linear equation \( y = mx + b \). The slope is expressed as a ratio \( \frac{\text{rise}}{\text{run}} \), indicating how much the line travels up or down (rise) for a given horizontal distance (run). In our example, the slope is \( \frac{2}{5} \). This means that for every 5 units we move to the right along the x-axis, the line rises 2 units up. Understanding the slope helps us generate additional points that lie on the line, facilitating an accurate graphing of the linear equation.

Plotting Points

Plotting points based on the slope and y-intercept allows us to visually represent a linear equation on a graph. To plot points:

• Begin at the y-intercept, such as (0,3) in our exercise.
• Use the slope \( \frac{2}{5} \) to move right 5 units (the "run") and up 2 units (the "rise").
• This takes us to the next point, which is (5,5).
• Mark each point clearly on the Cartesian plane.

By connecting these points with a straight line, this visualizes the equation, making it easier to analyze and understand the line's behavior.

Cartesian Plane

The Cartesian plane, or coordinate plane, is a two-dimensional graphing space divided by a horizontal x-axis and a vertical y-axis. It features four quadrants, allowing for the representation of points using ordered pairs (x, y). This system is fundamental for graphing all sorts of data, particularly linear equations. In our exercise, the Cartesian plane provides the framework for graphing the line. The x-axis represents horizontal movement, while the y-axis shows vertical changes. This grid is crucial for accurately plotting our y-intercept of (0,3) and illustrating the slope \( \frac{2}{5} \) by moving 5 units to the right and 2 units up from the starting point. Through the Cartesian plane, we can visually interpret relationships between equations and geometric space.