Problem 2 - 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 \(f(x)\). $$ f(x)=3-4 x $$

Step-by-Step Solution

Answer

The values are \( a = -4 \) and \( b = 3 \).

Step 1: Identify the standard form

A linear function is typically given in the standard form: \( f(x) = ax + b \). Our task is to identify the constants \( a \) and \( b \) from this equation.

Step 2: Rearrange the given function

The given function is \( f(x) = 3 - 4x \). To match the standard form \( f(x) = ax + b \), we rearrange it to: \( f(x) = -4x + 3 \).

Step 3: Match the standard form to the rearranged function

Compare the rearranged function \( f(x) = -4x + 3 \) with \( f(x) = ax + b \). This gives us \( a = -4 \) and \( b = 3 \).

Step 4: Confirm the values of a and b

After identifying \( a = -4 \) and \( b = 3 \), check if substituting these back into the standard form recreates \( f(x) = -4x + 3 \). It does, confirming our values are correct.

Key Concepts

Standard Form of a Linear Equation

In algebra, a linear function is a mathematical expression that describes a straight-line graph. The standard form for a linear equation is written as \( f(x) = ax + b \), where '\( a \)' is the coefficient of \( x \) and '\( b \)' is the constant. This formula is straightforward because it clearly separates the variable component from the constant.

• '\( a \)' determines the slope of the line, indicating how steep it is.
• '\( b \)' represents the y-intercept, showing where the line crosses the y-axis.

The role of each component in shaping the graph is essential for analyzing and interpreting linear functions. By structuring linear equations in this standard way, it’s easier to comprehend and manipulate the equation when solving problems. Understanding how to write and use the standard form is a fundamental skill in algebra.

Identifying Coefficients

Identifying coefficients in a linear equation means finding the numerical factors that multiply the variables in an equation.

For our example, the given function is \( f(x) = 3 - 4x \). To identify its coefficients, we compare it with the standard form \( f(x) = ax + b \). This helps us match the terms easily:

• Look at the term with \( x \). In \( -4x \), \( a=-4 \).
• The constant term without \( x \) is \( b = 3 \).

Identifying these coefficients helps us understand significant properties of the linear function. The coefficient of \( x \), known as the slope, affects the angle of the line. On the other hand, the constant term shows where the line crosses the y-axis.

Algebraic Manipulation

Algebraic manipulation involves rearranging an equation to simplify it or solve for variables. In our exercise, the given function \( f(x) = 3 - 4x \) needed to be rearranged to fit the standard form. This involved adjusting the terms to present the equation as \( f(x) = -4x + 3 \).

Here are the steps involved in this manipulation:

• Rearrange the equation so the \( x \) term appears first: swap \( 3 - 4x \) to \( -4x + 3 \).
• Compare with the standard form \( f(x) = ax + b \).
• Extract \( a \) and \( b \) by matching each component: \( a = -4 \) and \( b = 3 \).

Such algebraic manipulation allows for greater flexibility and is crucial for solving more complex algebraic problems. By rearranging equations, students can better understand how different components interact in a function.