Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

After creating a table of values for x from -3 to 3 and finding corresponding y values, plot those points onto a graph to get the linear line for the equation \( y = x - 2 \).

Step 1: Understanding the Equation

The equation given is \( y = x - 2 \). This is a linear equation which means the graph will be a straight line. The slope of the line will be '1'(coefficient of x) and the y-intercept will be '-2' (constant term).

Step 2: Create a Table

Next, you need to create a table of \(x\) and \(y\) values to get points on the graph. For each \(x\), calculate the corresponding \(y\) as \(y = x - 2 \). Thus, you will have 7 pairs of (x, y) values.

Step 3: Plotting the Graph

Plot each of these (x, y) pairs onto a graph, with x on the horizontal axis and y on the vertical axis. Since we have a straight line, two points would be enough to draw the line but use all to verify the correctness. Draw a line that passes through these points. This gives you the graph of the equation \( y = x - 2 \).

Key Concepts

Graphing

Graphing is an essential skill in mathematics that helps us visualize equations and understand their behavior. To graph an equation means to create a visual representation of the relationship between variables, typically on a coordinate plane. In this exercise, we are tasked with graphing the equation \( y = x - 2 \).

To start graphing, identify two axes: the horizontal axis (often the \(x\)-axis) and the vertical axis (often the \(y\)-axis). Every point on a graph is determined by an ordered pair \((x, y)\). You plot these points on the graph to visualize the equation.

For our equation, we need to calculate the \(y\) values for a given set of \(x\) values \([-3,-2,-1,0,1,2,3]\), plot these points on the graph, and connect them with a straight line. This visualization helps you understand the trend or pattern the equation represents.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to express linear equations conveniently and clearly. Typically, it is written as \( y = mx + b \), where \(m\) represents the slope, and \(b\) represents the y-intercept. This form is particularly useful in graphing because it quickly gives us crucial information about the line.

For the equation \( y = x - 2 \), we can identify that the slope \( m \) is 1, and the y-intercept \( b \) is -2. This means:

• The slope \( m = 1 \) tells us that for every unit increase in \(x\), \(y\) increases by 1 unit. The line rises at a 45-degree angle to the right.
• The y-intercept being \(-2\) indicates that the line crosses the y-axis at the point (0, -2).

Understanding the slope and y-intercept is crucial because they show how steep the line is and where it intersects the y-axis, respectively.

Plotting Points

Plotting points is a fundamental part of graphing linear equations. It involves finding specific coordinates that satisfy the equation and placing them accurately on the graph. By plotting points, you can visualize the exact path of the equation.

To plot the equation \( y = x - 2 \), you begin by selecting or being given specific \(x\) values to test. For each \(x\), compute \(y\) using the equation, which gives you a series of ordered pairs \((x, y)\). For example:

• \(x = -3\), \(y = x - 2 = -3 - 2 = -5\), so the point is \((-3, -5)\).
• \(x = 0\), \(y = x - 2 = 0 - 2 = -2\), so the point is \((0, -2)\).
• \(x = 3\), \(y = x - 2 = 3 - 2 = 1\), so the point is \((3, 1)\).

These points are plotted on the coordinate plane.

Once plotted, they help connect the dots with a straight line, demonstrating the line that the equation represents.

Straight Line Graph

A straight line graph is a type of graph that results when a linear equation is plotted. It visually shows a constant rate of change, which is a defining feature of linear equations.

For the equation \( y = x - 2 \), plotting the ordered pairs from the previous step results in a straight line. This line demonstrates a uniform increase in the \(y\) value as the \(x\) value increases.

Some key characteristics of the straight line graph include:

• It extends infinitely in both directions, represented by arrows on the ends of the line in a graph.
• It maintains a consistent slope. No matter where you measure along the line, the rate of change (slope) remains consistent.
• In this specific equation, the line passes through the y-intercept \(+/- 2\) and rises at an angle corresponding to the slope of 1.

Understanding straight line graphs helps in interpreting data, solving problems, and predicting future values in the field of mathematics.