Problem 24 - College Algebra with Modeling and Visualization

Question

State the slope of the graph of \(f\). Interpret this slope. $$ f(x)=6-x $$

Step-by-Step Solution

Answer

The slope is -1, indicating a negative relationship between \(x\) and \(f(x)\).

Step 1: Identify the function type

The given function is a linear function of the form \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Here, \(f(x) = 6 - x\) which can be re-written as \(f(x) = -x + 6\).

Step 2: Determine the slope

In the re-written equation \(f(x) = -x + 6\), the coefficient of \(x\) is \(-1\). Therefore, the slope \(m\) of the graph of \(f\) is \(-1\).

Step 3: Interpret the slope

The slope of \(-1\) means that for every unit increase in \(x\), the function \(f(x)\) decreases by 1 unit. This indicates a downward trend or a negative relationship between \(x\) and \(f(x)\).

Key Concepts

Understanding the Slope of a Line

The slope of a line is a fundamental concept when dealing with linear functions. It describes how one variable changes in response to another in a two-dimensional space. For any line represented algebraically by a linear equation of the form \(y = mx + c\), the slope \(m\) indicates the line's steepness and direction on a graph. In simple terms, it tells you how far and in which direction \(y\) changes as \(x\) increases or decreases.

In the example given, \(f(x) = 6 - x\), we can recognize the slope \(m\) from the equation \(-x + 6\) as being \(-1\). This negative slope means that for every 1 unit increase in \(x\), the value of \(f(x)\) decreases by 1 unit.

Key points to remember about slopes:

• A positive slope means an upward trend as \(x\) increases.
• A negative slope, like \(-1\), indicates a downward trend.
• A zero slope means the line is perfectly horizontal, showing no change.

Recognizing the slope helps in understanding the relationship between variables, which is essential in data interpretation and prediction.

Graph Interpretation of Linear Functions

Graph interpretation becomes easy when you grasp the idea of a linear function. A linear function like \(f(x) = 6 - x\) is visually represented by a straight line on a graph. The slope \(-1\) we've previously identified plays a crucial role in shaping this line.

Each point on the graph shows a specific \((x, f(x))\) coordinate. As you plot several points within the function, they will align to form a straight descending line from left to right due to the slope being \(-1\).

• The y-intercept is \(6\), the point where the line crosses the y-axis. It tells us the value of \(f(x)\) when \(x\) is zero.
• The slope dictates the angle of the line. Here, a slope of \(-1\) forms a -45 degree angle with the x-axis when plotted on standard graph paper.

Understanding the graphical representation helps in visualizing mathematical relationships and deriving logical conclusions regarding trends and behaviors of functions. This graph interpretation skill is invaluable for analyzing real-world data and forecasts.

Fundamentals of Function Analysis

Function analysis involves examining a mathematical function to understand its properties and implications. Let's take the function \(f(x) = 6 - x\). In simplest terms, analyzing this linear function involves identifying characteristics like slope, intercepts, and overall behavior of the function.

This function is foundational due to its simplicity and linear nature, which makes it an excellent starting point for analysis:

• **Linear properties:** It's a straight-line function, showing proportional changes between inputs (\(x\)) and outputs (\(f(x)\)).
• **Slope:** As previously mentioned, \(-1\) dictates a negative relationship, meaning as \(x\) increases, \(f(x)\) decreases.
• **Intercepts:**
  - **Y-intercept:** Occurs at \(6\), where \(x = 0\).
  - **X-intercept:** Achieved when \(f(x) = 0\), resulting in \(x = 6\), so the point \((6, 0)\) lies on the line.

By analyzing this function, one gains a clear insight into its linear characteristics and predictable behavior. This kind of analysis is crucial not just for mathematics but for fields ranging from physics to economics, where understanding how one variable affects another is key.