Problem 30 For each table, tell whether the... [FREE SOLUTION]

Chapter 4: Problem 30

For each table, tell whether the relationship between x and y could be linear, quadratic, or an inverse variation, and write an equation for the relationship. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} \\ \hline y & {0.25} & {1} & {2.25} & {4} & {6.25} \\ \hline\end{array} $$

Short Answer

Expert verified

The relationship is quadratic with equation: $ y = x^2 - 1 $.

Step by step solution

Identify the Relationship Type

Examine the values of $x$ and $y$ to ascertain whether the relationship is linear, quadratic, or inverse variation. Note any patterns in the changes of $y$ corresponding to changes in $x$. If the change in $y$ is proportional to $x$, it's likely linear; if $y$ changes with $x^2$, it might be quadratic; and if $y$ increases as $x$ decreases reciprocally, it could be inverse variation.

Compare Changes in $x$ and $y$

Calculate the differences in $x$ and $y$. For example: When $x$ changes from 1 to 2, $y$ changes from 0.25 to 1, so $\text{change in } y = 1 - 0.25 = 0.75$.When $x$ changes from 2 to 3, $y$ changes from 1 to 2.25, so $\text{change in } y = 2.25 - 1 = 1.25$, and continue this for other intervals.

Calculate Second Differences for Quadratic Relationship

Since first differences in $y$ are not constant, check for quadratic relationships by analyzing second differences. Second differences are calculated by taking the difference of the first differences.

Determine the Equation

From the second differences, if they are constant, it's a quadratic relationship. Assume the form $ y = ax^2 + bx + c $ and solve for $a$, $b$, and $c$ using the given points.

Write the Equation

Based on solving the equations formed in Step 4 from the given data points: For $x = 1, y = 0.25$, calculate constants that fit the quadratic model. Through comparison and solving for the variables in $y = ax^2 + bx + c$, the correct equation can be derived.

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Key Concepts

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

linear equations

Linear equations represent relationships where changes in one variable produce proportional changes in another. This means if you plot these relationships on a graph, they form a straight line. For instance, if we have a relationship described by the equation $ y = mx + b $, 'm' is the slope, and 'b' is the y-intercept. The slope 'm' tells us how much 'y' changes for every unit increase in 'x'. To identify a linear relationship from a table of values, we would check if the differences between the 'y' values are consistent as 'x' increases. If the differences (commonly known as 'first differences') are the same, it confirms a linear relationship.

quadratic equations

Quadratic equations are a step up in complexity from linear equations. These equations describe parabolic relationships, often represented in the form $ y = ax^2 + bx + c $. Here, 'a', 'b', and 'c' are constants, with 'a' determining the parabola's width and direction. If 'a' is positive, the parabola opens upwards, and if negative, it opens downwards. To identify a quadratic relationship from a data set, we check for constant second differences. First differences are derived by subtracting consecutive 'y' values, and if these first differences change at a constant rate, the second differences will be consistent. This consistency indicates a quadratic relationship.

inverse variation

Inverse variation is a unique type of relationship often contrasted with linear and quadratic relationships. In an inverse variation, the product of two variables remains constant. This means as one variable increases, the other decreases proportionally, which can be described by the equation $ y = \frac{k}{x} $, where 'k' is a constant. To determine if a relationship is inverse, you'd check if the multiplication of corresponding pairs of 'x' and 'y' is consistent across the data set. In a table, if multiplying each 'x' by its corresponding 'y' results in the same value for all pairs, the relationship is inverse.

mathematical patterns

Mathematical patterns are fundamental to understanding different types of equations and relationships. Recognizing patterns helps in categorizing relationships as linear, quadratic, or inverse. By systematically analyzing the changes in data, we can identify consistent trends that match known patterns. For instance, consistent first differences indicate linear patterns, while consistent second differences point towards quadratic patterns. Recognizing patterns requires practice, but it is a powerful tool in solving various algebraic problems and deducing the nature of relationships in data.

second differences

Second differences play a key role in identifying quadratic relationships. When examining a set of values, first differences are the changes between consecutive 'y' values. Second differences are then the changes between these first differences. If these second differences are constant, it implies that the data follows a quadratic pattern. This approach helps pinpoint the nature of a relationship when it鈥檚 not immediately clear from the raw data. Calculating second differences is essential for confirming quadratic equations and understanding the underlying patterns in the data.

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Short Answer

Step by step solution

Identify the Relationship Type

Compare Changes in \(x\) and \(y\)

Calculate Second Differences for Quadratic Relationship

Determine the Equation

Write the Equation

Key Concepts

linear equations

quadratic equations

inverse variation

mathematical patterns

second differences

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