Problem 22 - Algebra and Trigonometry

Question

A linear function is given.(a) Find the average rate of change of the function between \(x=a\) and \(x=a+h .\) (b) Show that the average rate of change is the same as the slope of the line. $$ g(x)=-4 x+2 $$

Step-by-Step Solution

Answer

The average rate of change is \(-4\), which is the same as the slope of the line.

Step 1: Determine function values

Start by finding the values of the function at the points of interest. Given the function \( g(x) = -4x + 2 \), find \( g(a) \) and \( g(a+h) \). Substitute \( x = a \) into the function to get \( g(a) = -4a + 2 \). Then substitute \( x = a + h \) into the function to get \( g(a+h) = -4(a+h) + 2 = -4a - 4h + 2 \).

Step 2: Calculate the average rate of change

The average rate of change of the function from \( x = a \) to \( x = a + h \) is given by \( \frac{g(a+h) - g(a)}{h} \). Substituting the values calculated in Step 1 gives: \[ \frac{(-4a - 4h + 2) - (-4a + 2)}{h} = \frac{-4a - 4h + 2 + 4a - 2}{h} = \frac{-4h}{h} = -4.\]

Step 3: Interpret the average rate

The calculation shows that the average rate of change is \(-4\). This value represents how much the function value changes for a unit increase in \(x\).

Step 4: Relate to the slope of the line

For a linear function in the form \( y = mx + b \), the slope \( m \) is the coefficient of \( x \). Here, the linear function is \( g(x) = -4x + 2 \), which means the slope \( m \) is \(-4\). This confirms that the average rate of change from \( x = a \) to \( x = a+h \) is equal to the slope of the line.

Key Concepts

Average Rate of Change

The average rate of change measures how a function's output values change relative to its input values over a specific interval. It's like finding out how fast something is changing on average over a journey. Here, we look at the function from two points, say from \( x = a \) to \( x = a + h \). To find this change, we take the difference in the function's values at these points and divide by the difference in the \( x \)-values. For example, consider the function \( g(x) = -4x + 2 \). At point \( x = a \), the function value is \( g(a) = -4a + 2 \). At point \( x = a + h \), it becomes \( g(a + h) = -4a - 4h + 2 \). The average rate of change is then calculated as: - \( \frac{g(a+h) - g(a)}{h} \)- Simplifying, we substitute values: \[ \frac{(-4a - 4h + 2) - (-4a + 2)}{h} \]- Which turns into: \[ \frac{-4h}{h} = -4 \]This value \(-4\) tells us that for every one unit increase in \(x\), \(g(x)\) decreases by 4 units, on average, over that interval.

Slope of a Line

The slope of a line is a fundamental aspect of linear equations. It tells us how steep or flat the line is. For any line expressed as \( y = mx + b \), the slope \( m \) is the coefficient of \( x \). The slope is a measure of change and reflects how one variable affects another.In our function \( g(x) = -4x + 2 \), the slope \( m \) is \(-4\). This negative sign indicates that as \( x \) increases, \( g(x) \) decreases, showing a downward or negative slope. Knowing the slope is like understanding the rules of a game. For instance:- A positive slope means the line goes uphill, from left to right.- A negative slope, like ours, means it goes downhill.- A zero slope indicates a flat line.Hence, when we relate this to the average rate of change, both values equate to \(-4\), confirming they describe the same rate of change per unit interval.

Function Evaluation

Function evaluation is the process of finding a function's value at a specific point by substituting the variable with the desired number. It’s like testing a recipe by using specific ingredients to see the final dish.In the exercise, we use function evaluation to find:- \( g(a) = -4a + 2 \): Here, we substitute \( x \) with \( a \) to compute the function's output.- \( g(a+h) = -4(a+h) + 2 = -4a - 4h + 2 \): To check the function at \( x = a + h \), we substitute and simplify to find the new function value.This evaluation enables us to compute the average rate of change, providing a solid understanding of how the function behaves over an interval. It’s important because it lets you explore how small changes in \( x \) affect \( g(x) \), helping to predict and understand the function's behavior effectively.