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Chapter 2: Problem 51

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

Short Answer

Expert verified
Identify constant slopes; if found, use them to find a linear equation.

Step by step solution

01

Analyzing the Table for Linear Relationships

To determine if a table represents a linear function, check if there is a constant rate of change between the x-values and y-values. This means the difference in y divided by the difference in x (i.e., the slope) should be constant for all pairs of consecutive data points in the table. For instance, if a table has points \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \), then the slope should be \( \frac{y_2 - y_1}{x_2 - x_1} = \frac{y_3 - y_2}{x_3 - x_2} \).
02

Calculating the Slope

Calculate the slope (m) using two consecutive data points from the table. For example, if the table gives the points (1, 2), (3, 6), then:\[m = \frac{6 - 2}{3 - 1} = \frac{4}{2} = 2\]This indicates that for each increase in x by 1 unit, y increases by 2 units.
03

Forming the Linear Equation

Once a constant slope is confirmed, use one of the points and the slope to write the equation of the line in the slope-intercept form: \( y = mx + b \). Substitute the slope and coordinates from one point to find the y-intercept (b). Using the point (1, 2) and slope (2):\[2 = 2 \cdot 1 + b \]\[b = 0\]The linear equation is \( y = 2x \).
04

Verifying the Equation

Verify the linear equation by checking all points in the table to ensure they satisfy the equation \( y = 2x \). Substituting the x-values from the table into the equation should yield the corresponding y-values as given in the table.

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Key Concepts

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

Slope Calculation
To calculate the slope, which is a key component of understanding linear functions, we need to find the rate at which the y-values change with respect to the x-values. This change is measured between two consecutive points on the graph of the function. The slope is expressed as the ratio of the difference in y-coordinates to the difference in x-coordinates. Mathematically, if you have two points,
  • Point 1: \(x_1, y_1\)
  • Point 2: \(x_2, y_2\)
the slope \(m\) is calculated as:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This computation gives you a numeric value that tells you how steep the line is. A positive slope means the line rises as x increases, a negative slope means it falls, and a slope of zero means the line is horizontal. If the slope is constant between consecutive points, it indicates that the function is linear.
Rate of Change
The rate of change is an important concept connected to slope in linear functions. It describes how a quantity changes over time. In the context of a linear function, this change is constant and is the same as the slope we calculate.
This constant rate of change indicates how much the dependent variable (often y) changes for a unit increase in the independent variable (often x). For example, if the slope \(m\) is 2, it means that when x increases by 1, y increases by 2.
  • Constant rate of change = Linearity
  • Rate of change = Slope = Steepness of Line
Understanding the rate of change helps in predicting future values and understanding the relationship between the two variables in practical situations, like speed or costs.
Slope-Intercept Form
When we express a linear equation, one of the most common forms used is the slope-intercept form. This form makes it easy to see both the slope and where the line crosses the y-axis. The general form of a linear equation in slope-intercept form is:\[y = mx + b\]where \(m\) represents the slope of the line, and \(b\) represents the y-intercept, the point where the line crosses the y-axis.
This form is particularly useful because:
  • \(m\) directly shows the rate of change.
  • \(b\) provides information on starting values.
To find these values, you calculate the slope as discussed, and solve for \(b\) using one point on the line. This makes predicting values and graphing the line straightforward.
Verifying Linear Equations
After finding an equation that seems to fit the data from a table, it鈥檚 important to verify its accuracy. Verification ensures that all data points in the table satisfy the equation you formed. This means substituting each x-value in the equation should yield the corresponding y-value in the table.
For example, if your equation is \( y = 2x \) and you have a point (1, 2) from your table:
  • Substitute x = 1 into the equation: \( y = 2(1) = 2 \)
  • The output matches the y-value (2) of the point (1, 2)
Continue this process for all points in your data set. If all points satisfy the equation, then the line accurately represents the data.

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Most popular questions from this chapter

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {4} & {5} & {6} & {7} & {8} & {9} & {10} & {11} & {12} & {13} \\ \hline y & {44.8} & {43.1} & {38.8} & {39} & {38} & {32.7} & {30.1} & {29.3} & {27} & {25.8} \\ \hline\end{array}$$

For the following exercises, use each set of data to calculate the regression line using a calculator or other technology tool, and determine the correlation coefficient to 3 decimal places of accuracy. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {100} & {80} & {60} & {55} & {40} & {20} \\ \hline y & {2000} & {1798} & {1589} & {1580} & {1390} & {1202} \\\ \hline\end{array}$$

The number of people afflicted with the common cold in the winter months dropped steadily by 25 each year since 2002 until 2012. In 2002, 8,040 people were inflicted. Find the linear function that models the number of people afflicted with the common cold \(C\) as a function of the year, \(t\). When will less than 6,000 people be afflicted?

For the following exercises, use the descriptions of the pairs of lines to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither? Line \(1 :\) Passes through \((8,-10)\) and \((0,-26)\) Line \(2 :\) Passes through \((2,5)\) and \((4,4)\)

When temperature is 0 degrees Celsius, the Fahrenheit temperature is \(32 .\) When the Celsius temperature is \(100,\) the corresponding Fahrenheit temperature is \(212 .\) Express the Fanrenheit temperature as a linear function of \(C,\) the Celsius temperature, \(F(C) .\) a. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius. b. Find and interpret \(F(28)\) . c. Find and interpret \(F(-40)\) .
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