Problem 15 - Algebra and Trigonometry

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$y=x-2$$

Step-by-Step Solution

Answer

The line graph for the equation \(y=x-2\) has its y-intercept at -2 and slope of 1. When the given x-values are substituted into the equation, the resulting coordinates for the line are \[(-3,-5), (-2,-4), (-1,-3), (0,-2), (1,-1), (2,0), (3,1)\].

Step 1: Substitute the Given x-Values

Plug each of the given x-values into the equation \(y=x-2\) and solve for y. The resulting ordered pairs are: \[(-3,-5), (-2,-4), (-1,-3), (0,-2), (1,-1), (2,0), (3,1)\]. These pairs will serve as the coordinates for the graph.

Step 2: Plot the Points

Plot the calculated pairs in a Cartesian coordinate system. Make sure each point corresponds to its respective coordinates.

Step 3: Draw the Line

Once all points are plotted, a pattern should emerge. The points should form a straight line, because the equation \(y=x-2\) represents a linear function. Carefully connect all points with a straight line and ensure that the line extends beyond the last point on both ends.

Key Concepts

Cartesian coordinate system

When graphing linear equations, we often rely on the Cartesian coordinate system, also known as the x-y plane. This coordinate system consists of two perpendicular lines, or axes. The horizontal line is called the x-axis, while the vertical line is called the y-axis. Where these two axes intersect is known as the origin, labeled as (0,0).

A point on this plane is represented as an ordered pair \(x, y\). The x-coordinate tells us how far to move horizontally from the origin, and the y-coordinate tells us how far to move vertically. With these coordinates, we can easily plot points and visualize relationships between variables. In our exercise, we use the Cartesian coordinate system to graph the linear equation by plotting and connecting ordered pairs.

Ordered pairs

Ordered pairs are fundamental in graphing equations on the Cartesian coordinate system. An ordered pair is expressed as \(x, y\), where x and y are coordinates on the x and y axes, respectively. The first number, x, designates the position horizontally, and the second number, y, specifies the position vertically.

For the linear equation \(y=x-2\), we generate ordered pairs by substituting given x-values into the equation to find the corresponding y-values. This gives us specific points like (-3, -5) and (0, -2). Such pairs are crucial as they provide the precise locations of points we need to plot on the graph.

Linear function

A linear function is a type of function where the relationship between the variables is a straight line when graphed. The general form of a linear equation is \(y=mx+b\), where m represents the slope of the line, and b is the y-intercept (the point where the line crosses the y-axis).

In our exercise, the linear equation \(y=x-2\) has a slope (m) of 1 and a y-intercept (b) of -2. This means that for every unit increase in x, y increases by 1, and the line crosses the y-axis at -2. Linear functions are straightforward to graph since they yield a constant rate of change, resulting in a straight line.

Equation solving

Solving equations involves finding the values of variables that make the equation true. In the exercise, we began with the linear equation \(y=x-2\) and solved it for specific x-values. This means substituting each x-value into the equation to find the corresponding y-value.

For instance, when x is -3, substituting gives \(y=(-3)-2 = -5\). Repeating this step for each x-value results in a list of ordered pairs, such as (-3, -5) and (2, 0), each representing a solution to the equation. This step is critical for plotting the points and understanding the graphical representation of the equation.