Problem 36 - Precalculus Essentials

Question

The table of values was generated by a graphing utility with a TABLE feature. Use the table to solve $$\begin{array}{|c|c|c|c|c|} \hline X & Y 1 & Y 2 & & \\ \hline-2 & 9 & 5 & & \\ -1 & 4 & 4 & & \\ 0 & 1 & 3 & & \\ 1 & 1 & 1 & & \\ 2 & 4 & 6 & & \\ 3 & 9 & -1 & & \\ 4 & 16 & -2 & & \\ 5 & 25 & -3 & & \\ 6 & 36 & -4 & & \\ \hline 7 & 49 & -5 & & \\ \hline X=-3 & & & & & \\ \hline \end{array}$$ Does the graph of \(Y_{1}\) pass through the origin?

Step-by-Step Solution

Answer

No, the graph of \(Y_{1}\) does not pass through the origin.

Step 1: Understand the Table

Look at the table and locate the X values that are given (in the first column) along with the corresponding Y1 values (in the second column).

Step 2: Identify the origin

Identify the origin as the point (0,0). This means that for a graph to pass through the origin, when X is 0, Y should also be 0.

Step 3: Check the Y1 value for X=0

Consider the row where X is 0. The Y1 value there is 1 which is not 0. Therefore, the graph of \(Y_{1}\) does not pass through the origin.

Key Concepts

Origin

The origin is a fundamental point in the coordinate system where the values of both the x-axis and y-axis are zero. In simpler terms, the origin is located where the horizontal line (x-axis) and vertical line (y-axis) intersect. This point is denoted as (0,0).

The origin serves as a reference point for locating other points on the graph.

• When discussing whether a graph passes through the origin, it means the graph must include the point (0,0).
• From a visual perspective, this ensures that if you were to draw the graph on a coordinate plane, it would touch the central point where both axes meet.

In the given problem's table, when x is 0, the corresponding y-value for Y1 is 1 and not 0. Therefore, the graph does not pass through the origin.

Table of Values

A table of values offers a structured way to represent data points that relate to functions and their respective y-values, given certain x-values. These tables provide a straightforward method for plotting graphs and analyzing the properties of functions.

Here's how a table of values is typically structured and used:

• The first column represents the x-values, which are inputs into the functions. These are sometimes known as independent variables.
• Each subsequent column represents the y-values (outputs) for particular functions corresponding to those x-values. The first function is labeled as Y1 and the second as Y2 in this table.

By looking at the table, you analyze how the function behaves. Understanding the relationship between these variables makes solving problems about the function's graph much easier. For instance, one can determine if the graph touches the origin by seeking the row where x is 0.

Coordinate System

A coordinate system is a two-dimensional number system where each point is defined by a pair of numbers known as coordinates. This system is the foundation of graphing and allows us to visualize mathematical concepts.

In the typical Cartesian coordinate system:

• The horizontal line is the x-axis, and the vertical line is the y-axis.
• Any point on the plane is identified by an ordered pair \((x,y)\), which signifies its position relative to the origin, which is the center point (0,0).

The coordinate system is critical in graphing functions by plotting points from the table of values. Each row from the table represents a point on this plane, where the x-value dictates position along the x-axis, and the y-value along the y-axis.

By plotting these points for Y1 from the table, you can draw the curve representing the function visually. The intersection of lines through these points reveals much about the function's nature, such as not passing through the origin, as seen in the provided example.