Problem 92 - College Algebra

Question

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. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 5 & 10 & 20 & 25 \\ \hline \boldsymbol{k}(\boldsymbol{x}) & 13 & 28 & 58 & 73 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Linear equation: \( y = 3x - 2 \)

Step 1: Determine Linear Relationship

To determine if the table represents a linear function, check if the rate of change between consecutive values of \( k(x) \) is constant. This means calculating the differences \( \Delta y \) for consecutive values of \( x \) and checking if they are equal.

Step 2: Calculate Differences

Calculate the difference between consecutive \( y \) values: \( \Delta y_{1} = 28 - 13 = 15 \) for \( x=5 \) to \( 10 \). \( \Delta y_{2} = 58 - 28 = 30 \) for \( x=10 \) to \( 20 \). \( \Delta y_{3} = 73 - 58 = 15 \) for \( x=20 \) to \( 25 \). Calculate corresponding \( \Delta x \) values: \( \Delta x_{1} = 10 - 5 = 5 \), \( \Delta x_{2} = 20 - 10 = 10 \), \( \Delta x_{3} = 25 - 20 = 5 \).

Step 3: Evaluate Constancy of Rate

Find the slopes \( m \) using the rate of change \( \frac{\Delta y}{\Delta x} \). Calculate: \( m_{1} = \frac{15}{5} = 3 \), \( m_{2} = \frac{30}{10} = 3 \), \( m_{3} = \frac{15}{5} = 3 \). All slopes are equal, indicating a constant rate of change.

Step 4: Derive Linear Equation

Since a linear relationship exists, derive a linear equation of the form \( y = mx + b \). With \( m = 3 \), substitute one point to find \( b \). Using \( x = 5 \), \( y = 13 \): \( 13 = 3(5) + b \rightarrow b = 13 - 15 = -2 \). Thus, the equation is \( y = 3x - 2 \).

Key Concepts

Rate of Change

When exploring a linear function, the term "Rate of Change" often comes up. This concept refers to how much one quantity changes, on average, relative to the change of another quantity. In the context of a linear function, this is evaluated by considering the differences between consecutive values of the outputs divided by the differences in the inputs.

For the given problem, we calculate the rate of change by determining the slope, represented by the formula \(m = \frac{\Delta y}{\Delta x}\). Here, \(\Delta y\) is the change in the output value, while \(\Delta x\) is the change in the input value.

• For \(x = 5\) to \(10\), \(\Delta y = 15\) and \(\Delta x = 5\), so \(m = 3\).
• For \(x = 10\) to \(20\), \(\Delta y = 30\) and \(\Delta x = 10\), so \(m = 3\).
• For \(x = 20\) to \(25\), \(\Delta y = 15\) and \(\Delta x = 5\), so \(m = 3\).

Seeing a consistent rate of change (all equal to 3) suggests that the function we are analyzing is indeed linear.

Linear Equation

A Linear Equation represents a relationship between two variables that creates a straight line when graphed on a coordinate plane. It is commonly written in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

In the exercise's solution, after confirming that the rate of change is constant, we determined that the function can be modeled by a linear equation. Using the slope \(m = 3\) calculated previously, substituting a known point (\(x = 5\), \(y = 13\)) helped us find the y-intercept \(b\).

• Setting up the equation based on \(y = mx + b\): \(13 = 3(5) + b\)
• Solving for \(b\): \(13 = 15 + b \implies b = -2\)

Thus, the derived linear equation for this function is \(y = 3x - 2\), effectively modeling the data highlighted in the table.

Table of Values

A Table of Values is a powerful tool for visualizing and verifying relationships within data, often used in identifying linear functions. By organizing input-output pairs, it becomes easier to assess whether a consistent pattern or relationship exists, such as a linear trend.

The values provided in a table format reveal the potential for linearity by allowing us to examine differences in output values relative to shifts in corresponding input values.

• The inputs in our example table are \(x = 5, 10, 20, 25\).
• Corresponding outputs are \(k(x) = 13, 28, 58, 73\).
• By calculating consecutive changes, you can examine potential consistency in changes, crucial for identifying a linear function.

This structured approach reveals that the outputs follow a consistent pattern relative to the changes in inputs, confirming a linear relationship is present in this set of values.