Problem 23 - College Algebra

Question

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

Step-by-Step Solution

Answer

After plotting the points, you should have a V-shaped graph moving upwards in a straight line from the point (0,1).

Step 1: Understanding the Equation

This equation represents the absolute value function, shifted upwards by one unit on the y-axis. An absolute value function transforms negative input values into positives, but keeps positive input values the same.

Step 2: Generate the Table of Values

For \( x = -3, -2, -1, 0, 1, 2, 3 \), calculate the corresponding \( y \) values by substituting \( x \) into the equation \( y = |x| + 1 \). This results in the following points: (-3,4), (-2,3), (-1,2), (0,1), (1,2), (2,3), (3,4).

Step 3: Plot the Points

Plot these points on a standard set of axes. Connect the points for a graphical representation of the equation.

Key Concepts

Graphing Equations

Graphing equations is a fundamental skill in algebra that involves drawing a visual representation of an equation on a coordinate plane. This process converts abstract mathematical expressions into a form that can be more easily understood and analyzed. By creating a graph, you gain insight into the properties of the function, such as its shape, intercepts, and areas where it increases or decreases.

Start with identifying the type of equation you are working with. In our case, the equation is of the absolute value type. You'll then choose a series of input (x) values, which are used to calculate the corresponding output (y) values. These x and y values are what you'll plot on the graph, helping to visualize the equation's behavior across a certain domain.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of the direction. For any real number x, the absolute value is denoted as \( |x| \) and is defined as:

• \( |x| = x \) if \( x \geq 0 \)
• \( |x| = -x \) if \( x < 0 \)

In graphing, the absolute value function tends to create a 'V' shape, with the point of the 'V' located at the origin if there is no horizontal or vertical shift. However, in the equation \( y = |x| + 1 \), we see that the entire function is shifted up by one unit along the y-axis. This shift moves the vertex of the 'V' from the origin to the point (0,1), affecting the position of the graph but not its characteristic shape.

Table of Values

A table of values is a practical tool used to organize the input-output relationship in a function or equation. It provides a clear view of how changes in the input value x lead to changes in the output value y. To create a table of values for the equation \( y = |x| + 1 \), assign x to various numbers (in this case, -3 to 3) and compute the corresponding y values. The table helps in predicting the pattern of the function and is an essential step before plotting points on the graph. With the points from the table, you can systematically plot each pair and begin to see the outline of the graph coming to life.

Plotting Points

Plotting points is the process of marking a dot at the coordinates that represent the x (horizontal value) and y (vertical value) of each pair on a grid, known as the Cartesian coordinate system. For the absolute value function \( y = |x| + 1 \), the table of values tells us exactly where to place these points. When plotted, they reveal the overall shape of the function. It is crucial to accurately plot these points because even a slight deviation can misrepresent the function's behavior. After plotting all the pairs from your table, connect them to form the graph of the equation. This graph can then be analyzed to understand various properties of the function, such as its continuity, symmetry, and intercepts with the axes.