Problem 158 - Algebra and Trigonometry

Question

Graph \(y=2 x\) and \(y=2 x+4\) in the same rectangular coordinate system. Select integers for \(x,\) starting with \(-2\) and ending with 2.

Step-by-Step Solution

Answer

The graph will have two lines. The line that corresponds to \(y=2x\) passes through the points (-2, -4), (-1, -2), (0, 0), (1, 2), and (2, 4), and the line that corresponds to \(y=2x+4\) passes through the points (-2, 0), (-1, 2), (0, 4), (1, 6), and (2, 8). The line for \(y=2x+4\) is 4 units above the line for \(y=2x\), indicating the vertical shift of the line.

Step 1: Identify the Equations

We have two linear equations that we need to plot on the same graph: \(y=2x\) and \(y=2x+4\).

Step 2: Identify the Range

The problem asks for all integer values for \(x\) from \(-2\) to 2, so we can make a table of x-values and solve each equation for \(y\).

Step 3: Calculate Y-values for the First Equation

We substitute the x-values into our first equation \(y=2x\), and solve for \(y\). This gives us the points (-2, -4), (-1, -2), (0, 0), (1, 2), and (2, 4).

Step 4: Calculate Y-values for the Second Equation

Similarly, substitute the x-values into our second equation \(y=2x+4\), and solve for \(y\). This gives us the points (-2, 0), (-1, 2), (0, 4), (1, 6), and (2, 8).

Step 5: Plot Points

Plot these points on the same graph, and label each point.

Step 6: Draw the Lines

Draw a line through the points for each equation. The line for \(y=2x\) will be lower than the line for \(y=2x+4\).

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional space defined by a pair of perpendicular axes. These axes are labeled as the x-axis (horizontal) and the y-axis (vertical). Each point within this plane is determined by an ordered pair, written as (x, y), which represents the coordinates of the point.

• The x-coordinate shows the position to the right (positive) or left (negative) of the vertical y-axis.
• The y-coordinate shows the position above (positive) or below (negative) the horizontal x-axis.

Through this system, it becomes clear and intuitive to represent geometric figures and algebraic equations visually. Graphing both lines and curves is possible by plotting specific points on the coordinate plane and drawing connections based on mathematical relationships.

Linear Equations

Linear equations represent relationships between two variables, generally x and y, in which their positions are proportional. A linear equation takes the form of y = mx + b, where:

• m is the slope, a measure of the steepness or inclination of the line.
• b is the y-intercept, which is where the line crosses the y-axis.

In the given exercise, two linear equations are considered: y = 2x and y = 2x + 4. The slope, m, for both is 2, which means both lines rise two units up for every one unit they move to the right. However, their y-intercepts differ due to the +4 in the second equation, shifting it upward by 4 units. As a result, while parallel, these lines do not overlap and are distinct when graphed on the same coordinate plane.

Coordinate Points

Coordinate points are essential in plotting graphs. Each point describes a location within the rectangular coordinate system using two values: the x-coordinate and y-coordinate. To effectively graph equations, especially linear ones, you often compute multiple coordinate points to establish the path of a line.
For the exercise provided, coordinate points were calculated and plotted for two equations, y = 2x and y = 2x + 4. Here is how the process was done:

• For y = 2x, points like (-2, -4), (-1, -2), and (2, 4) were determined. These are simply the result of substituting selected x-values in the equation to find corresponding y-values.
• For y = 2x + 4, after substituting the same x-values, points like (-2, 0), (0, 4), and (2, 8) were identified.

These coordinate points were then plotted on a graph, ensuring each pair correlates accurately in the Cartesian system, and connected to form distinct lines representing the equations. This plotting serves as a foundational skill for not just linear equations, but diverse mathematical functions, ensuring visual understanding of algebraic concepts.