Problem 31 - Thinking Mathematically

Question

Graph each equation in Exercises 21-32. Select integers for \(x\) from \(-3\) to 3 , inclusive. \(y=|x|+1\)

Step-by-Step Solution

Answer

The completed graph for the equation \(y = |x| + 1\) starts from the point (0,1) and forms a 'V' shape extending upwards at an angle of 45 degrees, with one arm in the first quadrant and the other in the second quadrant.

Step 1: Create a Table of Values

Choose values of \(x\) from -3 to 3 (inclusive) and calculate the corresponding values of \(y\) by substituting each value of \(x\) into the equation \(y = |x| + 1\). This will result in pairs of (x, y) values.

Step 2: Calculation of y Values

For each selected value of \(x\), \(y\) is calculated by applying the absolute value operation to \(x\) and adding 1. The absolute value of a number is simply its magnitude, and is always positive. Hence, for negative values of \(x\), we multiply by -1 to get the positive counterpart before adding 1. For positive \(x\) values, we simply add 1. For example, if \(x = -3\), \(y = |-3| + 1 = 3 + 1 = 4\). This process is repeated for all values of \(x\) to obtain all pairs of (x, y) points.

Step 3: Plot the Points

Finally, using the pairs of (x, y) values obtained, points are plotted on a graph. Each pair represents a point in the Cartesian coordinate system. A curve or line is then drawn connecting these points to form the graph of the equation \(y = |x| + 1\).

Key Concepts

Absolute Value

Understanding the absolute value in mathematics is crucial, especially when dealing with linear equations like the one given in the exercise. The absolute value of a number simply refers to its distance from zero on the number line, regardless of direction. Therefore, the absolute value of any number is always non-negative, emphasizing magnitude rather than direction.
For instance:

• The absolute value of \(-3\) is \(|-3| = 3\).
• The absolute value of \(2\) is \(|2| = 2\).

This idea is applied in the equation \(y = |x| + 1\), where regardless of whether \(x\) is positive or negative, \(y\) will be equal to the non-negative magnitude of \(x\) plus one. In this case, for each chosen integer from \(-3\) to \(3\), you calculate the absolute value and then add 1 to find the corresponding \(y\)-value. Understanding this concept helps simplify how we interpret and manipulate mathematical expressions involving absolute values.

Coordinate System

The coordinate system, also known as the Cartesian plane, is essential for graphing equations like \(y = |x| + 1\). A coordinate system consists of two perpendicular lines: the horizontal axis (x-axis) and the vertical axis (y-axis). Together, they form a grid that helps us visualize mathematical relationships through plotting points.

• Each point in this system has a pair of coordinates \( (x, y) \) indicating its position.
• The origin is where both axes meet, at the coordinate (0,0).
• Quadrants I, II, III, and IV divide the plane and each point lies in one of these quadrants based on its signs.

In our exercise, when you plot the given points, each (x, y) pair calculated from the equation reflects a position on this plane. Each point shows a relationship described by your equation, forming a `V` shape when connected. This framework is fundamental for interpreting and graphing functions accurately.

Table of Values

When graphing a linear equation involving absolute values, a table of values serves as a helpful tool. It provides a clear and organized way to keep track of values of \(x\) and their corresponding \(y\) values generated by the equation.For the given exercise, you would construct the table by:

• Choosing integer values for \(x\) from \(-3\) to \(3\).
• Calculating \(y\) for each chosen \(x\) by plugging it into the equation \(y = |x| + 1\).
• Recording each result as a pair \( (x, y) \).

For example, if you substitute \(x = -2\) into the equation, you calculate \(y = |-2| + 1 = 3\), yielding the point (-2, 3). These systematically listed points make it easier to plot each one on the coordinate plane accurately. This meticulous approach ensures that the graph reflects the equation correctly through the plotted path.