Problem 6 - 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 \(f(x)\). $$ f(x)=-6 $$

Step-by-Step Solution

Answer

\( a = 0 \) and \( b = -6 \).

Step 1: Understand the Given Function

The given function is a constant function, written as \( f(x) = -6 \). In this case, there is no \( x \) term in the function.

Step 2: Compare with the Linear Function Form

A linear function is generally written as \( f(x) = ax + b \). For the given function \( f(x) = -6 \), we need to compare it to get the values of \( a \) and \( b \). Since there is no \( x \) term, the coefficient \( a = 0 \).

Step 3: Identify "b" in the Function

The constant part of the function, \( -6 \), can be compared to the \( b \) in the standard form \( ax + b \). This gives us \( b = -6 \).

Step 4: Conclude the Values of 'a' and 'b'

Concluding from the steps above, the linear function can be rewritten as \( f(x) = 0 \cdot x - 6 \). Therefore, \( a = 0 \) and \( b = -6 \).

Key Concepts

Constant Function

A constant function is a unique type of function where the output value remains the same, no matter what the input is. In simpler terms, no matter what value you plug into the function, the answer will always be the constant that defines the function. For example, with the constant function given as \( f(x) = -6 \), the output will always be \(-6\), regardless of the value of \(x\).
This characteristic is due to the lack of a variable \(x\) term, resulting in the variable practically "disappearing" when it comes to its influence on the function's value.
The mathematical form of a constant function fits perfectly into the general equation for a linear function \( ax + b \) by assigning \( a = 0 \) and using the constant value for \( b \). Hence, a constant function like \(-6\) can be seen as \(0x - 6\).

• Constant functions are horizontal lines on a graph.
• They have a slope of zero, as there's no change when \( x \) changes.

Linear Equations

Linear equations form a fundamental part of algebra, characterized by expressions that graph as straight lines. The general form is \( ax + b = c \). These equations can be simplified to the popular form \( f(x) = ax + b \).
The linear equation encompasses variations based on the values of \( a \) and \( b \).
If \( a = 0 \), like in a constant function, the line will be horizontal.
When working with linear equations, the slope of the line is a key aspect. It's represented by \( a \) in the equation \( ax + b \). If \( a \) changes, it affects the steepness of the line: the larger the \( a \), the steeper the slope.

• Linear equations can model relationships with constant rates of change.
• The y-intercept of the line is given by \( b \), which is where the line crosses the y-axis.

Algebra

Algebra involves using symbols, typically letters, to represent numbers and express mathematical relationships. It provides the tools to solve equations, such as linear functions, by manipulating expressions to find unknowns.
In our context of a linear function \( f(x) = ax + b \), algebra helps to deduce values like \( a \) and \( b \) from specific cases or to generalize them.

• Substitution is a common algebraic technique, where you replace a variable in an equation with another value to simplify the problem-solving process.
• Balancing equations, a core algebra method, ensures both sides of the equation remain equal during manipulations.
• Basic algebraic skills are crucial for interpreting and solving linear equations and understanding function behaviors, like constant functions.

By relying on rules such as the distributive property or combining like terms, algebra simplifies complex expressions, making it easier to work with diverse mathematical scenarios.