Problem 24 - 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 given linear equation \(y = x + 2\) can be visualized in a 2D graph, by plotting the points obtained from choosing values of \(x\) from \(-3\) to \(3\) and calculating the corresponding \(y\) values. The coordinates to plot are (-3,-1), (-2,0), (-1,1), (0,2), (1,3), (2,4), (3,5). Joining these points will give a straight line which is the graphical representation of the equation.

Step 1: Determine the coordinates

Since the task is to select integer values of \(x\) from \(-3\) to \(3\), substitute these values into the equation \(y = x + 2\) to compute the corresponding \(y\) values. This will give us the coordinates we need for our graph.

Step 2: Calculate the coordinates

The coordinates using the equation \(y = x + 2\) will be: For \(x = -3\), \(y = -3 + 2 = -1\) giving (-3,-1) For \(x = -2\), \(y = -2 + 2 = 0\) giving (-2,0) For \(x = -1\), \(y = -1 + 2 = 1\) giving (-1,1) For \(x = 0\), \(y = 0 + 2 = 2\) giving (0,2) For \(x = 1\), \(y = 1 + 2 = 3\) giving (1,3) For \(x = 2\), \(y = 2 + 2 = 4\) giving (2,4) For \(x = 3\), \(y = 3 + 2 = 5\) giving (3,5)

Step 3: Plot the coordinates

Draw a graph and mark out the coordinates (-3,-1), (-2,0), (-1,1), (0,2), (1,3), (2,4), and (3,5). Join these points to make a straight line. This line is the graphical representation of the equation \(y = x + 2\).

Key Concepts

Graphing Coordinates

Graphing coordinates is a foundational step in expressing mathematical relationships visually. When dealing with linear equations, like our example equation \(y = x + 2\), each unique solution can be represented as a point on a graph. This point is defined by a pair of numbers, known as coordinates, that specify its position on a Cartesian plane.
To find the coordinates:

• Choose values for \(x\).
• Substitute these values into the equation to find the corresponding \(y\) values.
• Pair these \(x\) and \(y\) values to get coordinates, e.g., (-3,-1), (0,2), etc.

In a graph, the first number of the pair (\(x\)) represents the horizontal position, while the second number (\(y\)) indicates the vertical position. Mastering how to plot these coordinates means you can graphically solve equations and visually analyze their behavior.

Integer Substitution

Integer substitution is a simple yet powerful technique in solving equations graphically. It involves replacing a variable, typically \(x\), with specific integer values to find corresponding solutions. In our example, we replace \(x\) with integers from -3 to 3.
Why use integer values? Because they are straightforward and provide consistent results, which simplifies the graphing process. Integer values create distinct, easy-to-plot points.
Substituting these values into \(y = x + 2\) gives a set of specific points — (-3,-1), (-2,0), and so forth. By limiting \(x\) to these integers, we simplify the graphing process and obtain a span of the equation’s behavior across a small interval. This method can be applied to any linear equation to determine clearer graphical representations.

Plotting Graphs

Plotting graphs is the culmination of graphing techniques. It allows us to transform abstract equations into visual stories. To plot the graph of \(y = x + 2\), follow these steps:

• Draw a Cartesian coordinate system with horizontal (x-axis) and vertical (y-axis) lines intersecting at the origin (0,0).
• Use the coordinates derived from integer substitutions, such as (-3,-1) and (3,5), to locate points on the graph.
• Once all points are plotted, connect them with a straight line. Since this is a linear equation, the graph is a straight line extending in both directions.

The resulting graph shows the relationship the equation describes. In this example, it demonstrates how \(y\) consistently increases as \(x\) does. Practicing plotting makes reading and understanding equations much more intuitive.