Problem 1 - 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 \(f(x)\). $$ f(x)=5-2 x $$

Step-by-Step Solution

Answer

\(a = -2\), \(b = 5\).

Step 1: Identify the standard form of the linear function

A linear function can be written in the standard form as \(f(x) = ax + b\), where \(a\) is the coefficient of \(x\) and \(b\) is the constant term.

Step 2: Rewrite the given function

The given function is \(f(x) = 5 - 2x\). We will rewrite this in the standard form \(f(x) = ax + b\) for better comparison.

Step 3: Rearrange the terms to match the standard form

Reorder \(5 - 2x\) to read \(-2x + 5\). This allows us to easily compare with \(ax + b\).

Step 4: Identify \(a\)

The expression \(-2x\) shows that the coefficient of \(x\), which is \(a\), is \(-2\).

Step 5: Identify \(b\)

The constant term in \(f(x) = -2x + 5\) is \(5\). Therefore, \(b = 5\).

Key Concepts

Coefficient

In a linear function, the **coefficient** is the number multiplied by the variable, usually represented by the letter \( x \) in our equation. Consider the standard linear form \( f(x) = ax + b \). Here, \( a \) is the coefficient of \( x \). It's a crucial part of the function, as it determines the slope of the line.

• If \( a \) is positive, the slope goes upward from left to right. This means the line is rising.
• If \( a \) is negative, the slope declines, causing the line to fall from left to right.

Understanding the coefficient helps in predicting how the line moves across a graph. Specifically, the magnitude of \( a \) represents how steep the line is.
For the given function \( f(x) = -2x + 5 \):

• The coefficient \( a \) is \(-2\). This tells us that the slope is negative, which makes the line fall, making the line steeper as \( x \) increases.

Knowing this is fundamental when graphing linear equations, as it helps us picture the line's direction and steepness.

Constant Term

The **constant term** in a linear function gives the value of the function when the input variable \( x \) is zero. In the standard form \( f(x) = ax + b \), the constant term is \( b \). You can think of it as the starting point of the line on the graph, where it crosses the y-axis.

• No matter what the coefficient is, the constant term tells you where the line will intercept the y-axis by setting \( x = 0 \).
• It serves as the initial value of the function before any changes due to \( x \) are applied.

Knowing the constant term also provides an immediate and valuable insight into the graph of the linear function.
In the function \( f(x) = -2x + 5 \):

• The constant term \( b \) is \( 5 \), meaning the line intersects the y-axis at the point (0, 5).

This value offers a clear starting point when plotting the graph and can often simplify understanding the linear relationship.

Standard Form

The **standard form** of a linear function is a conventional way to write the equation, typically expressed as \( f(x) = ax + b \). This format not only helps in clearly identifying the coefficient and constant term, but it also simplifies the comparison of different linear functions.

• By writing the function in standard form, one can easily spot the values of \( a \) and \( b \), as it separates them distinctly.
• The order where \( x \) comes first provides a consistent method of visualizing and calculating linear equations.

Recasting a linear equation into standard form can also facilitate easier graphing and understanding of its properties.
For the example given, \( f(x) = 5 - 2x \), rearranging it to standard form results in \( f(x) = -2x + 5 \).

• This makes the identification of the coefficient \( a = -2 \) and the constant term \( b = 5 \) straightforward.

Overall, utilizing the standard form for linear equations enhances clarity, which benefits both basic calculations and visual representations on graphs.