Problem 4 - College Algebra with Modeling and Visualization

Question

A linear function \(f\) can be written in the form \(f(x)=a x+b .\) Identify a and b for the given \(f(x)\). $$ f(x)=10 x $$

Step-by-Step Solution

Answer

\( a = 10 \), \( b = 0 \)

Step 1: Understand the Form of f(x)

The given function is expressed as \( f(x) = 10x \). Compare this with the standard form of a linear function, \( f(x) = ax+b \).

Step 2: Identify the Value of a

In \( f(x) = 10x \), compare with \( f(x) = ax + b \): \( a = 10 \) because it is the coefficient of \( x \).

Step 3: Identify the Value of b

Notice that \( f(x) = 10x \) can be seen as \( f(x) = 10x + 0 \). Thus, \( b = 0 \) because it is the constant term added to \( ax \).

Key Concepts

Coefficient Identification

When dealing with linear functions, the goal is to identify the coefficients in the equation. Coefficients are the numbers that multiply the variables. For the given function, which is written in the form \( f(x) = 10x \), the standard format we compare against is \( f(x) = ax + b \). Here, \( a \) is the coefficient of \( x \), and \( b \) is the constant term without any variable. By comparing the two forms, we see:

• \( a = 10 \) because it multiplies \( x \).
• \( b = 0 \) because there is no constant term explicitly shown; it's like adding zero.

Identifying coefficients is a key skill in analyzing and understanding linear functions.

Standard Form

Understanding the standard form of a linear function is crucial for analyzing functions and solving equations. The standard form for linear equations is \( f(x) = ax + b \). This is where:

• \( a \) is the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, representing where the line crosses the y-axis.

For the function \( f(x) = 10x \), we compare it with the standard form. Our function doesn’t have a visible constant, which makes it seem like \( b = 0 \). This means the line passes through the origin of the graph. Recognizing this form helps in plotting the line and understanding its behavior.

Function Analysis

Analyzing a function involves looking at its components and understanding what they mean for the graph and its real-world applications. In our function \( f(x) = 10x \):

• The slope \( a = 10 \) means that for every unit increase in \( x \), \( f(x) \) increases by 10 units. This indicates a steep line.
• Since \( b = 0 \), the line goes through the origin \((0,0)\).

This analysis gives us a picture of how the function behaves: it is a straight line with a positive slope and no offset on the y-axis. Realizing this assists in predicting values and understanding the linear relationship in practical scenarios. Linearity implies constant rates of change, which are easy to work with in modeling symmetric relationships.