Problem 94 - College Algebra

Question

For the following exercises, which of the tables could represent function? For each that could be linear, find a linear equation that modees the data. $$ \begin{array}{|c|c|c|c|c|}\hline x & {2} & {4} & {8} & {10} \\ \hline h(x) & {13} & {23} & {43} & {53} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

The table is a function and linear with equation \(h(x) = 5x + 3\).

Step 1: Determine if the Table Represents a Function

To check if the table represents a function, confirm that for each unique input \(x\) value, there is exactly one output \(h(x)\) value. The table lists \(x\) values of 2, 4, 8, and 10 with corresponding \(h(x)\) values of 13, 23, 43, and 53. Since each \(x\) value has one specific \(h(x)\) value, this table represents a function.

Step 2: Check for a Constant Rate of Change

A linear function has a constant rate of change (constant slope). Calculate the rate of change between each pair of consecutive points: \((4, 23) - (2, 13)\) gives \((23-13)/(4-2) = 5\); \((8, 43) - (4, 23)\) gives \((43-23)/(8-4) = 5\); \((10, 53) - (8, 43)\) gives \((53-43)/(10-8) = 5\). Since the rate of change is constant, the function is linear.

Step 3: Find the Linear Equation

The slope \(m\) is 5, from the constant rate of change. Use one of the points, say \((2, 13)\), to find the y-intercept \(b\) using the formula \(h(x) = mx + b\):\[ 13 = 5(2) + b \]\[ 13 = 10 + b \]\[ b = 3 \]Thus, the linear equation is \(h(x) = 5x + 3\).

Key Concepts

Linear Equations

Linear equations are foundational in mathematics, representing relationships that maintain a constant rate of change. These equations are often expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. A unique feature of linear equations is that they graph as straight lines. The consistency of their rate of change makes them predictable and easy to interpret.
In the context of this exercise, checking whether a table could represent a function involves ensuring each input \(x\) has a unique output \(h(x)\). A linear equation emerges if the change between outputs remains consistent for equal changes in inputs. This ensures the function is a straight line, confirming its linearity.

Rate of Change

The rate of change in a function measures how much one quantity changes in relation to the change in another. For linear equations, this is often referred to as the slope. This tells us how steep the line is and the direction in which it moves.
To check if a function is linear, you calculate the rate of change between each consecutive pair of points. In the exercise, we see this as:

• Between \((2, 13)\) and \((4, 23)\), rate = \((23-13) / (4-2) = 5\)
• Between \((4, 23)\) and \((8, 43)\), rate = \((43-23) / (8-4) = 5\)
• Between \((8, 43)\) and \((10, 53)\), rate = \((53-43) / (10-8) = 5\)

A consistent rate of 5 indicates that the function is linear, confirming it follows a straight path on a graph.

Slope-Intercept Form

The slope-intercept form of a linear equation, given as \(y = mx + b\), is a straightforward way to model linear relationships. Here:

• \(m\) represents the slope, indicating the rate of change or how steep the line is.
• \(b\) is the y-intercept, which tells us where the line crosses the y-axis.

In the original exercise, identifying the slope was the key step. Using the constant rate of change calculated (\(m = 5\)) and any given point, such as \((2, 13)\), we can find the y-intercept. By substituting these values into the equation \(h(x) = mx + b\), we solve for \(b\) and get:\[13 = 5(2) + b\]\[13 = 10 + b\]\[b = 3\]Thus, the linear equation modeling this function is \(h(x) = 5x + 3\). This encapsulates the entire relationship between \(x\) and \(h(x)\) in a simple mathematical function.