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Chapter 7: Problem 51

Determine whether the values in each table could represent a linear relationship, a quadratic relationship, or neither. Explain your answers. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {-3} & {-2} & {-1} & {0} & {1} & {2} & {3} \\ \hline y & {-12.6} & {-9.2} & {-5.8} & {-2.4} & {1} & {4.4} & {7.8} \\ \hline\end{array} $$

Short Answer

Expert verified
Linear relationship, first differences are constant.

Step by step solution

01

Determine if the relationship is linear

To check if the relationship is linear, compute the differences between successive y-values. If these differences are constant, the relationship is linear.
02

Calculate first differences

Calculate the differences between each consecutive pair of y-values:y_{-2} - y_{-3} = -9.2 - (-12.6) = 3.4y_{-1} - y_{-2} = -5.8 - (-9.2) = 3.4y_{0} - y_{-1} = -2.4 - (-5.8) = 3.4y_{1} - y_{0} = 1 - (-2.4) = 3.4y_{2} - y_{1} = 4.4 - 1 = 3.4y_{3} - y_{2} = 7.8 - 4.4 = 3.4All differences are equal, showing a constant first difference.
03

Conclusion

Since the first differences are constant, the values in the table represent a linear relationship.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations form the backbone of many algebraic concepts. These equations are of the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the given exercise, the constant first differences between successive y-values indicate a linear relationship, meaning they fit the equation of a line. This is because a linear relationship maintains a consistent rate of change, represented by the slope in the equation. Recognizing linear equations helps simplify understanding algebraic relationships and solving problems efficiently. Whenever you encounter a set of values, checking for linearity can often provide a straightforward path to the solution.
Differences in Sequences
When analyzing sequences, especially in tables, calculating differences between consecutive values can reveal patterns. In the exercise, the first differences between successive y-values were calculated as 3.4, consistently. This consistency indicates that the sequence grows at a constant rate.

By finding these differences, which are known as first differences, algebra students can identify the type of relationship in the data. If the first differences are equal, the relationship is linear. If not, further analysis is needed to check for quadratic or other types of relationships. Understanding these differences is crucial in identifying linear relationships easily.
Graphing Linear Functions
Graphing linear functions visualizes the relationship between variables, making it easier to understand. With equations like \(y = mx + b\), plotting on a graph highlights the straight line formed by the relationship.

In the exercise, if you plot the given points on a graph with \(x\) on the horizontal axis and \(y\) on the vertical axis, you will see a straight line. This line validates the constant first differences calculated, reinforcing the linear relationship. Graphing these functions helps students see the continuity and consistency in data trends, allowing for better comprehension of linear relationships in algebra.
Algebraic Problem Solving
Algebraic problem solving involves a logical approach to identifying and solving equations. By recognizing patterns and relationships, as demonstrated in the original exercise, students can solve for unknowns and verify their relationships. The process of calculating differences, identifying consistencies, and graphing data are steps in this broader approach.

Effective problem solving in algebra means breaking down the problem into manageable steps. Start by identifying the type of equation, compute differences, graph the values if necessary, and make conclusions based on your analysis. These practices help ensure accurate and efficient solutions in algebra.

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