Problem 25 - College Algebra with Modeling and Visualization

Question

State the slope of the graph of \(f\). Interpret this slope. $$ f(x)=-\frac{3}{4} x $$

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{4}\), representing a downward trend.

Step 1: Identify the Equation Form

The given function is in the form of a linear equation, which is generally written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Extract the Slope

In the equation \( f(x) = -\frac{3}{4}x \), compare it with the standard form \( y = mx + b \). Here, the slope \( m \) is \( -\frac{3}{4} \).

Step 3: Interpret the Slope

The slope of \(-\frac{3}{4}\) means that for every 1 unit increase in \( x \), the value of \( f(x) \) (or \( y \)) decreases by \( \frac{3}{4} \) units. This indicates a downward trend or negative correlation in the graph.

Key Concepts

Slope Interpretation

The slope of a linear function provides a lot of insight into how the graph behaves. When we refer to the slope as seen in the equation \( f(x) = -\frac{3}{4}x \), it tells us that for every increase of 1 unit in the x-direction, the y-value, or \( f(x) \), decreases by \( \frac{3}{4} \).

This type of slope is an example of a negative slope, meaning the line slants downwards from left to right.

The concept of slope is crucial because it gives us a quick understanding of the rate of change. In simple terms, it tells us how steep the line is and in which direction it moves. With a negative slope such as \(-\frac{3}{4}\), think of the graph as a line sliding down a hill — it represents a decrease in the y-value as the x-value increases.

Linear Equations

Linear equations are fundamental in algebra and depict a straight line when graphed. At the heart of any linear equation is the general formula: \( y = mx + b \). Here, \( m \) represents the slope, while \( b \) is the y-intercept. This equation provides a straightforward way to model a variety of real-world situations.

In our equation, \( f(x) = -\frac{3}{4}x \), you notice that there is no '+ b' term. This absence means the y-intercept is 0, indicating the line crosses the origin (0,0).

Linear equations are powerful tools in both mathematics and applied sciences because they help in predicting values, identifying correlations, and making quick calculations. Imagine them as the building blocks for more complex mathematical frameworks. Understanding the components \( m \) and \( b \) allows you to graph any linear equation swiftly.

Negative Slope

A negative slope is an intriguing characteristic of linear functions. When you encounter a negative slope like \(-\frac{3}{4}\) from our equation \( f(x) = -\frac{3}{4}x \), it implies an inverse relationship between the variables \( x \) and \( f(x) \).

Here's what you should know about negative slopes:

• A negative slope slopes downward when moving from left to right across the graph.
• It indicates that as one variable increases, the other decreases. This is called a negative correlation.
• Negative slopes are common in scenarios where one factor diminishes against the increase of another, like the decrease in speed as resistance increases.

Understanding a negative slope helps in grasping how variables are interconnected inversely, paving the way for insights into areas such as economics, physics, and general trends of data representation.