Problem 27 - College Algebra

Question

For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points. $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{g}(\boldsymbol{x}) & -3.25 & 2 & 7.25 & 12.5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The table represents a linear function, \( g(x) = 5.25x - 8.5 \).

Step 1: Analyze Differences for Linearity

First, check if the function could be linear by calculating the differences between successive outputs \( g(x) \). Compute \( g(2) - g(1), g(3) - g(2), \) and \( g(4) - g(3) \). If these differences are constant, the function is linear.

Step 2: Calculate Differences

Calculate the differences: \( 2 - (-3.25) = 5.25 \), \( 7.25 - 2 = 5.25 \), and \( 12.5 - 7.25 = 5.25 \). Since all differences are the same, the function is linear.

Step 3: Determine Linear Function Equation

With constant differences, the function is linear. Use the slope-intercept form \( g(x) = mx + b \). Here, the slope \( m = 5.25 \). Find \( b \) using the point (1, -3.25): \( -3.25 = 5.25(1) + b \), solve for \( b \) to get \( b = -8.5 \).

Step 4: Construct Function

With \( m = 5.25 \) and \( b = -8.5 \), the linear function is \( g(x) = 5.25x - 8.5 \).

Key Concepts

Function Analysis

Function analysis is an important step in understanding how a function behaves. Before determining the type of function represented by a set of data points, we must observe the relationships between these points.
By analyzing the differences in output values, we can identify patterns and discern whether a function might be linear, exponential, or neither.

• For a linear function, the difference between successive outputs (also known as first differences) is constant.
• For an exponential function, the ratio between successive outputs should be constant.

In the given problem, we begin by checking the differences between consecutive values of the output function, \(g(x)\). Calculating each difference helps confirm whether the function has a constant rate of change, a hallmark of linearity. In this case, analyzing the differences revealed a consistent pattern, which suggested that the function would be linear.

Slope-Intercept Form

The slope-intercept form is a straightforward way to express a linear function. It makes it easy to identify key components, like the slope and the y-intercept.
This form is denoted as \(y = mx + b\), where:

• \(m\) represents the slope of the line, which is the consistent amount that \(y\) increases or decreases as \(x\) goes up by 1.
• \(b\) is the y-intercept, which is the value of \(y\) when \(x = 0\).

In our exercise, once we confirmed that the function is linear, we used the values to substitute into this format. The calculated slope, \(m = 5.25\), shows how much \(g(x)\) increases for every additional unit of \(x\). Finally, the y-intercept is found by inserting one data point into the equation form. This allows us to derive the complete equation, \(g(x) = 5.25x - 8.5\), directly revealing the function's behavior.

Determining Linear Equations

Determining linear equations involves finding both the slope and y-intercept, which define the behavior of the function. To determine the linear equation accurately, it is vital to follow these steps methodically:

1. Check for constant differences: Make sure the differences between successive outputs are equal, signaling linearity.
2. Calculate the slope \(m\): The constant difference itself is the slope of the line.
3. Identify a point: Use one of the data points in the format \((x_1, y_1)\).
4. Solve for the y-intercept \(b\): Insert the slope and your selected point into the slope-intercept formula \(y = mx + b\) and solve for \(b\).

In our situation, we utilized these logical steps by computing differences, identifying the slope as 5.25, and using the point (1, -3.25) to solve for the y-intercept, \(b = -8.5\).
This thorough approach always results in a complete understanding of how the linear function is structured and behaves across its domain.