Problem 43 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line satisfying the given conditions. assing through \((4,3)\) with 0 slope

Step-by-Step Solution

Answer

The line is horizontal through \(4,3\), with the equation \(y = 3\).

Step 1: Understanding the Slope

The slope of a line measures how steep the line is. A slope of 0 indicates that the line is perfectly horizontal.

Step 2: Identify the Line's Equation

For a line with a slope of 0, the equation is of the form \(y = c\). Here, \(c\) is the y-coordinate of any point on the line. Since the line passes through \(4,3\), \(c = 3\). Thus, the equation is \(y = 3\).

Step 3: Sketching the Graph

To sketch the graph, plot the point \(4,3\). Then, draw a horizontal line passing through this point. Every point on this horizontal line will have a y-coordinate of 3, confirming the equation \(y = 3\).

Key Concepts

Slope

The slope of a line tells us how steep the line is or how it inclines or declines. It measures the change in the y-coordinate for a change in the x-coordinate. Mathematically, the slope is represented as \(m = \frac{\text{rise}}{\text{run}}\). Here, 'rise' is the change in y, and 'run' is the change in x. For a horizontal line, the slope is 0, which means there is no vertical change as you move along the line. The line remains flat without any tilt or angle. Understanding the slope is crucial because it allows you to determine the nature and direction of a line on a graph.

Equation of a Line

The equation of a line is a mathematical expression showing the relationship between the x and y coordinates of every point on the line. For a horizontal line, the equation is always of the form \(y = c\), where \(c\) is a constant representing the y-coordinate of any point on the line. In simple terms, no matter what x value you choose, y will always be constant. In our example, since the line passes through the point \(4,3\), the line's equation is \(y = 3\). This tells us that every point on this line will have a y-coordinate of 3.

Graph Sketching

Sketching a graph helps visualize the relationship between variables. To sketch the graph of a horizontal line, follow these simple steps:

• Identify at least one point through which the line passes.
• Since the slope is 0, draw a straight horizontal line passing through this point.

In our example, you need to plot the point \(4,3\) first. Then, draw a horizontal line through this point. Remember, this line is parallel to the x-axis because it has a 0 slope, indicating there is no vertical change.

Plotting Points

Plotting points correctly is essential for accurate graphing. A point is represented as \( (x, y) \) where x is the horizontal coordinate and y is the vertical coordinate.
To plot the point \(4,3\):

• Start from the origin \(0,0\) on the graph.
• Move 4 units to the right along the x-axis.
• From there, move 3 units up along the y-axis.
• Mark this location as the point \((4,3)\).

Once you have this point on the graph, you can easily draw the horizontal line through it to complete your sketch.