Problem 32 - Algebra and Trigonometry

Question

Linear Functions Have Constant Rate of Change If \(f(x)=m x+b\) is a linear function, then the average rate of change of \(f\) between any two real numbers \(x_{1}\) and \(x_{2}\) is average rate of change \(=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}\) Calculate this average rate of change to show that it is the same as the slope \(m .\)

Step-by-Step Solution

Answer

The average rate of change is equal to the slope \( m \).

Step 1: Identify the function values

We start by finding the values of the function at both points, \( f(x_1) \) and \( f(x_2) \). For the linear function \( f(x) = mx + b \), this gives us \( f(x_1) = m \cdot x_1 + b \) and \( f(x_2) = m \cdot x_2 + b \).

Step 2: Substitute into the average rate of change formula

The formula for the average rate of change between two points \( x_1 \) and \( x_2 \) is given by \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \). Substituting our expressions for \( f(x_1) \) and \( f(x_2) \), we have:\[\frac{(m \cdot x_2 + b) - (m \cdot x_1 + b)}{x_2 - x_1}.\]

Step 3: Simplify the expression

Subtract \( m \cdot x_1 + b \) from \( m \cdot x_2 + b \). The \( b \) terms cancel out:\[\frac{m \cdot x_2 - m \cdot x_1}{x_2 - x_1}.\]

Step 4: Factor out the common factor

Notice that \( m \) is a common factor in the numerator, so factor it out:\[\frac{m(x_2 - x_1)}{x_2 - x_1}.\]

Step 5: Cancel the common terms

The \( x_2 - x_1 \) in the numerator and denominator cancel each other out, leaving:\[m.\]

Step 6: Conclusion

Thus, the average rate of change of the function \( f(x) = mx + b \) between any two points is simply \( m \), which is exactly the slope of the line.

Key Concepts

Constant Rate of Change

Linear functions are quite unique because they exhibit a constant rate of change across their entire domain. This means that no matter which two points you choose to evaluate on the function, the rate of change will always be the same. This is because the graph of a linear function is a straight line, and straight lines don't bend or curve, offering consistency in how they increase or decrease.
The constant rate of change is what makes linear relationships predictable. For instance, in a scenario where you're paid by the hour, your wages increase at a constant rate as your hours worked increase. This simple idea helps us model and understand various real-world situations where change happens consistently.
- Linear functions: consistent growth or decline - Example of hourly wages - Helps predict outcomes easily

Slope of a Line

The slope of a line is a fundamental concept that describes the steepness or direction of a line. In a linear equation like \( f(x) = mx + b \), \( m \) represents the slope. Mathematically, the slope is defined as the ratio of the "rise" (vertical change) to the "run" (horizontal change) between two distinct points on the line.
Understanding slope is crucial because it tells us how much the function value changes as the input changes. Positive slopes indicate an upward trend, whereas negative slopes indicate downward trends. A zero slope means the line is perfectly horizontal, showing no change between any two points.
- Ratio of vertical change to horizontal change- Indicators of line direction: positive, negative, or zero- Essential for graphing and comparing linear functions

Average Rate of Change

The average rate of change is a broader concept in mathematics that applies to various functions, not just linear ones. It reflects how much a function's output changes, on average, between two input values. For linear functions, the average rate of change matches the slope, since the change is constant.
The average rate of change formula is \( \frac{f(x_2) - f(x_1)}{x_2 - x_1} \), which works by finding the difference in function values between two points and dividing by the difference in the input values. Although other functions may have varying rates of change, for linear functions, this calculation will always simplify to the line's slope, \( m \).
- Applies to all functions but constant in linear- On a straight line, equals the slope- Key for evaluating changes between two values