Problem 18 - College Algebra

Question

Graph each equation .Let $x=-3,-2,-1,0,1,2, and 3. $$ y=2 x-4 $$

Step-by-Step Solution

Answer

The graph of the equation \(y = 2x - 4\) is a straight line that passes through the corresponding \(y\) values found by substituting \(x = -3,-2,-1,0,1,2,3\) into the equation.

Step 1: Generate the table of values

Substitute the given values of \(x\) into the equation \(y = 2x - 4\) to get corresponding \(y\) values. This will give a set of coordinates that can be plotted on the graph.

Step 2: Plot the coordinates

Next, plot the coordinates on a graph and, if possible draw a straight line through them. This line represents the graph of the equation \(y = 2x - 4\).

Step 3: Interpret the graph

The line is a graphical representation of all the solutions to the equation. Any point on this line will satisfy the equation \(y = 2x - 4\).

Key Concepts

Graphing Linear Equations

Graphing linear equations involves representing an equation on a coordinate plane as a straight line. This process makes it easy to see all possible solutions to the equation. When you graph a linear equation like \(y = 2x - 4\), you start by choosing some values for \(x\), such as \(-3, -2, -1, 0, 1, 2,\) and \(3\).
Substitute each \(x\) value into the equation to find the corresponding \(y\) values, generating points like \((-3, -10), (-2, -8),\) and so on.
The next step is to plot these points on the coordinate plane. Once plotted, a straight line is drawn connecting them to illustrate the graph of the equation.

• Visual representation: A straight line showing every possible solution.
• Slope indicates direction: The line's slope informs whether the line goes up or down as you move from left to right.
• Intercepts: Where the line crosses the axes provides additional insight into the equation.

Coordinate System

A coordinate system is a framework for defining locations on a plane using two numerical values, namely the x-coordinate and y-coordinate. This is referred to as the Cartesian coordinate system.
The coordinate plane is formed by two perpendicular lines called axes, which intersect at a point called the origin. The horizontal line is the x-axis, and the vertical line is the y-axis. Each point on the plane is defined by a pair of numbers called coordinates, written as \((x, y)\).

• Origin: The point \((0, 0)\), where the x-axis and y-axis intersect.
• Quadrants: The plane is divided into four regions known as quadrants.
• Positive/negative axes: Moving right/upwards from the origin increases values while left/downwards decreases them.

Understanding the coordinate system is crucial for graphing equations because it supplies the framework for plotting points and interpreting graphs.

Slope-Intercept Form

The slope-intercept form is a way of writing linear equations as \(y = mx + b\). This format is especially helpful in graphing and understanding linear equations.
In this formula:

• \(m\) represents the slope of the line. It indicates how steep the line is and the direction it tilts.
• \(b\) is the y-intercept. This is where the line crosses the y-axis when \(x = 0\).

For instance, in the equation \(y = 2x - 4\), the slope \(m\) is \(2\). This means for every unit increase in \(x\), \(y\) increases by \(2\). The y-intercept \(b\) is \(-4\), indicating the line crosses the y-axis at the point (0, -4).
The slope and y-intercept are key in constructing the graph of the equation, as they define both the orientation and position of the line relative to the axes.