Problem 35 - 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 \(Y_{2}\) pass through the origin?

Step-by-Step Solution

Answer

No, the graph of \(Y_{2}\) does not pass through the origin because the \(Y_{2}\) value for X=0 is 3, not 0.

Step 1: Identify the Corresponding Value

Find the row of the table where X equals zero.

Step 2: Check the \(Y_{2}\) Value

Find the corresponding \(Y_{2}\) value in that row.

Step 3: Determine if the Graph Passes Through the Origin

If the \(Y_{2}\) value for X equals 0 is not 0, then the graph does not pass through the origin. If the \(Y_{2}\) value for X equals 0 is in fact 0, then the graph passes through the origin.

Key Concepts

Graphing Utility

A graphing utility is a technological tool that can be extremely helpful when studying linear equations. Think of it as a digital assistant that helps visualize mathematical concepts. Among its many features, one of the most utilized is the TABLE feature. This function allows the user to input a function and obtain a corresponding table of values automatically, which saves time and minimizes computational errors.

When learning to graph linear equations, a graphing utility is particularly beneficial. It helps students to quickly see the relationship between variables by generating graphs from equations or vice versa. Stepping beyond the basic uses, advanced utilities also offer options for tracing along the curve, analyzing slopes, and even finding intersections with other lines or axes. By integrating this technology into practice, students can explore complexities and patterns within linear equations with greater ease and accuracy.

Table of Values

A table of values is a systematic way to organize data that shows how two variables relate to each other. It's particularly useful in graphing linear equations, where each row of the table represents a specific input-output pair. For example, in a linear equation with two variables, x and y, each row would display an x value and the corresponding y value calculated from the equation.

To create a table of values manually, choose a range of x-values – often including both negative and positive numbers – and then apply the equation to find each y-value. However, when using a graphing utility, this process is automated.

Constructing or analyzing tables of values can help improve understanding of the nature of a linear equation. Patterns become evident, like equal increments in x leading to consistent changes in y, which indicates a steady rate of change – a hallmark of linear relationships.

Origin of a Graph

The origin of a graph is the point where the x and y axes intersect, which is represented by the coordinates (0,0). In the context of graphing linear equations, the origin holds a unique position—it is the point of equilibrium where both variables have zero value.

When analyzing whether a linear graph passes through the origin, you look at the table of values, specifically when x is 0. If the corresponding y-value is also 0, then indeed, the graph intersects the origin. This is important because if a linear equation's graph passes through the origin, the equation is directly proportional, which means one variable is a constant multiple of the other.

Remember, a graph can cross the y-axis at points other than the origin. The specific y-intercept provides information about the equation's vertical shift from the origin and is key in graphing and understanding the equation's behavior.