Problem 88 - College Algebra with Modeling and Visualization

Question

If \(f(x)=a x+b,\) what does the difference quotient for function \(f\) equal? Explain your reasoning.

Step-by-Step Solution

Answer

The difference quotient equals \(a\).

Step 1: Understand the Definition

The difference quotient for a function \(f(x)\) is given by the formula \(\frac{f(x+h) - f(x)}{h}\), where \(h\) is a small increment in \(x\). It represents the average rate of change of the function over the interval \([x, x+h]\).

Step 2: Substitute into the Function

Substitute \(x + h\) into the function \(f(x) = ax + b\), so that \(f(x + h) = a(x + h) + b\). Simplifying this, we get \(f(x + h) = ax + ah + b\).

Step 3: Compute the Difference

Calculate \(f(x+h) - f(x)\). From Step 2, we have \(f(x+h) = ax + ah + b\) and \(f(x) = ax + b\). The difference is \((ax + ah + b) - (ax + b) = ah\).

Step 4: Form the Difference Quotient

Substitute the result from Step 3 into the difference quotient formula: \(\frac{f(x+h) - f(x)}{h}\). Therefore, \(\frac{ah}{h} = a\).

Step 5: Simplify the Expression

The \(h\) in the numerator and the \(h\) in the denominator cancel out, leaving \(a\) as the result. This shows that the difference quotient \(\frac{f(x+h) - f(x)}{h}\) simplifies to the constant \(a\), which is the slope of the linear function \(f(x) = ax + b\).

Key Concepts

Linear Function

A linear function is a type of function that creates a straight line when graphed on a coordinate plane. These functions are often written in the form \(f(x) = ax + b\), where \(a\) and \(b\) are constants. Here, \(a\) is the slope of the line and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
Linear functions have several characteristics:

• Consistency: They have a constant rate of change, meaning the slope \(a\) does not vary.
• Straight Line: When plotted, they form a straight line.
• Predictable Behavior: Because they form a straight line, predicting values within their range is straightforward using their equation.

Linear functions are foundational in understanding more complex types of functions and equations. This simplicity in structure makes learning about linear functions an essential part of developing math skills.

Average Rate of Change

The average rate of change of a function is a measure of how much the function's output value changes, on average, over a specified interval. For any function \(f(x)\), the average rate of change between two points \(x\) and \(x + h\) can be calculated using the difference quotient \(\frac{f(x + h) - f(x)}{h}\).
This concept is particularly important for understanding how a function behaves over an interval:

• Linear Functions: In linear functions like those expressed as \(f(x) = ax + b\), the average rate of change simplifies to the slope \(a\), regardless of the interval.
• Interpreting Change: If the average rate of change is positive, the function is increasing over that interval. If it's negative, the function decreases.
• Practical Applications: Measuring how things change, such as speed or growth rates, often involves finding the average rate of change.

By understanding the average rate of change, students can analyze and predict behaviors within different mathematical or real-world situations.

Slope

The slope of a line is a number that describes both the direction and the steepness of the line. In the context of linear functions, represented as \(f(x) = ax + b\), the slope \(a\) is a crucial part that defines the line's characteristic.
Here's why slope is indispensable:

• Direction: If \(a > 0\), the line increases as \(x\) increases. If \(a < 0\), the line decreases as \(x\) increases. A 0 slope indicates a horizontal line.
• Steepness: The greater the absolute value of \(a\), the steeper the line. A larger value for \(a\) means the line rises or falls more quickly.
• Application: Slope is widely used in fields like physics for velocity, economics for understanding changes in cost, and many other areas that involve rate of change.

Understanding slope, particularly within the scope of linear functions, helps students make sense of changes and trends, whether plotted graphically or applied in various practical contexts.