Problem 40 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The values of \(x\) for which \(Y_1 = Y_2\) are \(x = -1\) and \(x = 1\).

Step 1: Identify Field of Operation

You are working with a two-dimensional table that charts the values of \(x\), \(Y_1 (x)\) and \(Y_2 (x)\). You're asked to identify for which values of \(x\), \(Y_1\) equals \(Y_2\).

Step 2: Scan the Tables

By scanning through the table, you can find the values of \(x\) for which \(Y_1\) equals \(Y_2\). This involves comparing the \(Y_1\) and \(Y_2\) columns for each \(x\) value.

Step 3: Identify Matching Values

When scanning through the table, you see that at \(x = -1\) and \(x = 1\), \(Y_1\) equals \(Y_2\). Both pairs are (4,4) and (1,1) respectively.

Key Concepts

Two-Dimensional Table

A two-dimensional table is a simple yet powerful tool used to organize data that relies on two variables. In this context, the table arranges values based on the variables \( X \) and the corresponding outcomes of \( Y_1 \) and \( Y_2 \).
Understanding a two-dimensional table:

• It consists of rows and columns, where each row represents a different value of \( X \), and columns relate to different functions or outcomes, like \( Y_1 \) and \( Y_2 \).
• The headers often denote what the values are related to, for instance, "\( X \)" or "\( Y_1 \)" and "\( Y_2 \)" in this case.
• Such tables allow you to compare how two different functions behave for the same independent variable \( X \).
• It's vital for visualizing patterns or solving equations directly from the data given.

By leveraging the structured format of a two-dimensional table, one can efficiently pinpoint the relationships or specific points of intersection between different sets of data, as required in solving the exercise at hand.

Graphing Utility

A graphing utility is a technological tool designed to help visualize equations and data to gain better insights. These are often found in graphing calculators or computer software and play a vital role in managing numeric and algebraic computations automatically.
Key aspects of using a graphing utility:

• They create plots or graphs that offer a visual representation of mathematical functions based on input data or equations.
• Graphing utilities help generate tables of values that can highlight critical points where variables or functions meet desired criteria (e.g., intersections).
• These tools save time by quickly computing values without manual calculations for every point, especially beneficial when dealing with complex functions.

Utilizing a graphing utility's "TABLE" feature can be immensely helpful as it allows users to explore and dissect different outcomes quickly, assisting in visually interpreting where, for example, \( Y_1 = Y_2 \) occurs.

Comparative Analysis

Comparative analysis involves examining two or more items, variables, or datasets to understand their differences and similarities. In mathematical contexts, like the given exercise, it's crucial for highlighting where the values from two functions coincide.
Methods of conducting a comparative analysis:

• Start by listing the datasets or outputs in an organized manner, often using tables to clarify relationships.
• Analyze the data row by row, comparing each value of \( Y_1 \) to corresponding values of \( Y_2 \) for identical \( X \) values.
• Point out any \( X \) values where the outputs are equal, as they reflect points where the equations or functions intersect.

This type of analysis is critical in the textbook problem, where the task is to identify \( X \) values at which \( Y_1 = Y_2 \). By systematically comparing these values, students can discover these intersections effectively.

Equation Solving

Equation solving is the process of finding the values of variables that satisfy a given equation. It involves manipulating equations to isolate the variable of interest, making it central to understanding where two expressions yield the same result.
Approaches to solving equations using tables:

• Use the organized table to directly observe where the values of \( Y_1 \) and \( Y_2 \) align, implying solutions to the equation \( Y_1 = Y_2 \).
• Note the specific \( X \) values that produce matching outputs for both equations.
• Verify these solutions by considering the context of the problem and any additional constraints or patterns.

In this exercise, solving the equation was straightforward with the provided table, as it showcased exactly where the two expressions were equivalent at \( X = -1 \) and \( X = 1 \). This process provides insights into understanding algebraic concepts through real numerical examples.