Problem 23 - Thinking Mathematically

Question

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=x-2\)

Step-by-Step Solution

Answer

The graph of the equation \(y=x-2\) is a straight line with a slope of 1 and cuts the y-axis at -2. The points from -3 to 3 on the graph are (-3,-5), (-2,-4), (-1,-3), (0,-2), (1,-1), (2,0), (3,1).

Step 1: Identify the slope and y-intercept

From the equation we can see that the slope \(m\) is 1 and the y-intercept \(b\) is -2. Slope is the rise/run or the steepness of the line. The y-intercept is where the line crosses the y-axis.

Step 2: Create a table of values

We will create a table of values for \(x=-3\) to 3 and for each of these \(x\)-values, we’ll calculate the corresponding \(y\)-value using the formula \(y=x-2\). Hence, a list of coordinates (\(x\), \(y\)) will be obtained.

Step 3: Plot the points and draw the line

Plot each point from the table on the graph. After plotting the points, draw a straight line that best fits these points, extending the line beyond the plotted points, as needed. The line represents all the solutions of the equation.

Key Concepts

Slope

The slope of a linear equation tells you how steep the line is on the graph. It's often represented by the letter "m" and is found in the equation of a line in the form: \( y = mx + b \). Here, \( m \) is the slope. In simple terms, slope refers to how much a line rises or falls as it advances from left to right. To understand it better, consider slope as "rise over run."

• Rise: The vertical change, or how much the line goes up or down.
• Run: The horizontal change, or how far the line goes from left to right.

For example, a slope of 1, as in the equation \( y = x - 2 \), indicates that for every unit it runs (moves horizontally), it rises one unit (moves vertically). This implies a perfect 45-degree angle upwards. To find the slope in any equation, look at the coefficient in front of \( x \).

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of the equation \( y = mx + b \), the "b" represents the y-intercept. This means, it's the value of \( y \) when \( x = 0 \).

• In \( y = x - 2 \), the y-intercept is \( -2 \). This tells us that when \( x \) is zero, \( y \) will be \( -2 \). On the graph, this corresponds to the point (0, -2).

Y-intercepts are important because they offer a starting point for graphing a line. Finding this point first makes it easier to draw the line accurately using the slope from there.

Plotting points

Plotting points involves marking specific positions on the graph, using coordinates derived from the equation. This is an essential step in graphing because plotted points show where the line should actually go.
To plot points for the equation \( y = x - 2 \), you would follow these steps:

• Select values for \( x \), typically a mix of positive, negative, and zero should suffice, like from \( -3 \) to \( 3 \).
• Calculate the corresponding \( y \) values using the equation. For example, for \( x = -3 \), calculate \( y = -3 - 2 = -5 \). This works out to the point \( (-3, -5) \).
• Mark these points on the coordinate plane. Add all the points together to confidently position the line.

Plotting several points ensures that the line is consistent and true to the equation since it confirms the relationship given in \( y = x - 2 \).

Coordinate plane

The coordinate plane is a two-dimensional surface where all points are plotted using a pair of numbers, namely the x-coordinate and the y-coordinate. These numbers collectively determine the position of any point on the plane.

• X-axis: The horizontal line on the graph. It divides the plane into the upper and lower halves.
• Y-axis: The vertical line on the graph. It splits the plane into the left and right halves.

The point where these axes intersect is the origin, denoted by \( (0,0) \). The coordinate plane is essential for graphing not only linear equations but many other kinds of mathematical functions as well. For our line from \( y = x - 2 \), we plot several points based on calculated coordinates and draw the line through them, showing the behavior of the equation visually on this grid.