Problem 18 - Applied Calculus

Question

Find all the tables that have the given characteristic. A. $$\begin{array}{c|c|c|c|c} x & 0 & 40 & 80 & 160 \\ \hline y & 2.2 & 2.2 & 2.2 & 2.2 \\ \hline \end{array}$$ B. $$\begin{array}{c|c|c|c|c} \hline x & -8 & -4 & 0 & 8 \\ \hline y & 51 & 62 & 73 & 95 \\ \hline \end{array}$$ C.$$\begin{array}{c|c|c|c|c} \hline x & -4 & -3 & 4 & 6 \\ \hline y & 18 & 0 & 4.5 & -2.25 \\ \hline \end{array}$$ D. $$\begin{array}{c|c|c|c|c} \hline x & 3 & 4 & 5 & 6 \\ \hline y & 18 & 9 & 4.5 & 2.25 \\ \hline \end{array}$$ \(y\) could be a linear function of \(x\)

Step-by-Step Solution

Answer

Tables A and B are linear.

Step 1: Interpret the characteristic

The characteristic given is that \(y\) could be a linear function of \(x\). In mathematical terms, a relationship between \(x\) and \(y\) is linear if it can be described using the equation \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We will assess each table to determine if the data fits a linear pattern.

Step 2: Analyze Table A

For Table A:\[\begin{array}{c|c|c|c|c}x & 0 & 40 & 80 & 160 \y & 2.2 & 2.2 & 2.2 & 2.2 \\end{array}\]The \(y\)-values are constant. This represents a special case of a linear function where the slope \(m = 0\) and the equation is \(y = c\). Therefore, Table A is linear.

Step 3: Analyze Table B

For Table B:\[\begin{array}{c|c|c|c|c}x & -8 & -4 & 0 & 8 \y & 51 & 62 & 73 & 95 \\end{array}\]Calculate the slope \(m\) between each pair of points:- Between \((-8, 51)\) and \((-4, 62)\): \(m = \frac{62 - 51}{-4 - (-8)} = \frac{11}{4}\)- Between \((-4, 62)\) and \((0, 73)\): \(m = \frac{73 - 62}{0 - (-4)} = \frac{11}{4}\)- Between \((0, 73)\) and \((8, 95)\): \(m = \frac{95 - 73}{8 - 0} = \frac{22}{8} = \frac{11}{4}\)Since the slope is constant, the relation is linear. Thus, Table B is linear.

Step 4: Analyze Table C

For Table C:\[\begin{array}{c|c|c|c|c}x & -4 & -3 & 4 & 6 \y & 18 & 0 & 4.5 & -2.25 \\end{array}\]Calculate the slope \(m\) between each pair of points:- Between \((-4, 18)\) and \((-3, 0)\): \(m = \frac{0 - 18}{-3 - (-4)} = -18\)- Between \((-3, 0)\) and \((4, 4.5)\): \(m = \frac{4.5 - 0}{4 - (-3)} \approx 0.643\)- Between \((4, 4.5)\) and \((6, -2.25)\): \(m = \frac{-2.25 - 4.5}{6 - 4} = -3.375\)The slopes are not constant, indicating Table C is not linear.

Step 5: Analyze Table D

For Table D:\[\begin{array}{c|c|c|c|c}x & 3 & 4 & 5 & 6 \y & 18 & 9 & 4.5 & 2.25 \\end{array}\]Calculate the slope \(m\) between each pair of points:- Between \((3, 18)\) and \((4, 9)\): \(m = \frac{9 - 18}{4 - 3} = -9\)- Between \((4, 9)\) and \((5, 4.5)\): \(m = \frac{4.5 - 9}{5 - 4} = -4.5\)- Between \((5, 4.5)\) and \((6, 2.25)\): \(m = \frac{2.25 - 4.5}{6 - 5} = -2.25\)The slope is not constant, so Table D is not linear.

Key Concepts

Linear Equation

A linear equation is a mathematical expression that defines a linear relationship between two variables, typically "x" and "y". The standard form of a linear equation is given by \( y = mx + c \), where \( m \) represents the slope and \( c \) is the y-intercept. This form indicates a straight line when plotted on a graph, because it describes a line with constant slope and intercept from the y-axis.

In the context of our exercise, we are examining the tabulated values to see if a linear equation can describe them. By doing this, we essentially look for a pattern or relationship where changes in "x" always result in consistent changes in "y", thus forming a straight line when connected. Linear equations are foundational in algebra because they provide a simple way to connect and analyze real-world data that behaves linearly.

To verify if the relationship in data is linear, you need to check whether the change in \( y \) with respect to \( x \) remains constant. This constant rate of change fundamentally defines whether or not the table of values can be described by a linear equation.

Slope Calculation

The slope of a line in a linear equation is an important concept, as it defines the angle and direction of the line. The slope, denoted by \( m \), is calculated as the ratio of the change in \( y \) (vertical change) to the change in \( x \) (horizontal change), and is given by the formula \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \).

• A positive slope indicates that the line is increasing, slanting upwards as you move from left to right.
• A negative slope means that the line is decreasing, moving downwards as you move from left to right.
• A zero slope represents a horizontal line, indicative of constant \( y \) values.

In our exercise, assessing the slopes between the different points in the tables helps us determine linearity. A consistent slope between different pairs of points confirms that the data aligns with a straight line (linear relationship).

For instance, in Table A, the constant \( y \) values made the slope zero, which signifies a horizontal line and confirms linearity. Similarly, in Table B, the slope was consistently \( \frac{11}{4} \) across different points, again confirming a linear relationship.

Mathematical Analysis

Mathematical analysis involves closely examining the given data to determine if there are any recognizable patterns or characteristics that fit the criteria for a certain type of mathematical relationship. In this example, we sought to analyze tables to confirm if any contained data that could be described using a linear function.

For Tables C and D, we conducted a detailed analysis by calculating the slopes between various combinations of points. In Table C, different slopes such as \(-18\), \(0.643\), and \(-3.375\) showed inconsistency, therefore indicating the absence of a linear relationship. The varying slopes hinted at changing rates of increase or decrease in \( y \) for changes in \( x \).

• Inconsistent slopes suggest a non-linear pattern.
• Consistent slopes confirm linearity.

These analyses illustrate how mathematical analysis helps identify not only linear relationships but also deviations from linearity. Through methods such as slope calculation and graphical representation, one can efficiently visualize and confirm the nature of the data to accurately interpret it in terms of linear equations.