Problem 5 - 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 \(f(x)\). $$ f(x)=7 $$

Step-by-Step Solution

Answer

\( a = 0 \), \( b = 7 \)

Step 1: Understand the form of a linear function

A linear function is expressed as \( f(x) = ax + b \), where \( a \) is the slope and \( b \) is the y-intercept.

Step 2: Analyze the given function

The function given is \( f(x) = 7 \). This means the output is constant regardless of \( x \) value. It indicates that there is no dependence on \( x \), suggesting \( a = 0 \).

Step 3: Determine the values of \( a \) and \( b \)

Since \( f(x) = 7 \) is a horizontal line, this implies \( a = 0 \) and the constant term \( b = 7 \). The function can be rewritten as \( f(x) = 0 \cdot x + 7 \).

Step 4: Conclusion

For the function \( f(x) = 7 \), \( a = 0 \) and \( b = 7 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express linear functions. It is written as \( f(x) = ax + b \), where:

• \( a \) is the slope of the line, indicating its steepness and direction.
• \( b \) is the y-intercept, representing where the line crosses the y-axis.

This formula allows you to easily understand and visualize the behavior of a line.
For example, if \( a \) is positive, the line inclines upward, while a negative \( a \) indicates a downward slope.
The y-intercept \( b \) provides a starting point on the y-axis, from where you can apply the slope to find other points on the line.
By using this form, linear equations become more transparent, making graph interpretation straightforward.

Horizontal Lines

Horizontal lines are unique in that they remain perfectly flat, running parallel to the x-axis. In terms of a linear equation:

• The slope \( a \) is zero, meaning there is no vertical change as you move along the line.
• The function is expressed in the form \( f(x) = b \), indicating the constant y-value throughout.

These lines reflect a scenario where the output of the function remains unchanged, regardless of the input.
This characteristic is what defines a horizontal line as a form of a constant function. Understanding horizontal lines aids in grasping the concept of functions where output remains unaffected by changes in input.

Constant Functions

A constant function is a simple concept in mathematics where every output value is identical, no matter the input. Its general form is given by \( f(x) = b \), where:

• It lacks an \( x \) component that would affect the function's output.
• Every point on the graph of the function is aligned on the same horizontal line.

This absence of an \( x \) term means the slope \( a \) is zero, leading to a stable, unchanging horizontal line at \( y = b \).
In the example \( f(x) = 7 \), no matter what \( x \) value you choose, \( f(x) \) always equals 7.
Understanding constant functions strengthens the understanding of cases where a variable does not impact the outcome, emphasizing the importance of both slope and linear behavior in more complex linear function scenarios.