Problem 36 - Precalculus Essentials

Question

A formula in the form \(y=m x+b\) models the average retail price, \(y,\) of a new car \(x\) years after \(2000 .\) Would you expect \(m\) to be positive, negative, or zero? Explain your answer.

Step-by-Step Solution

Answer

The expected value for \(m\) would be positive. This is because as the years after 2000, represented by \(x\), increase, the average retail price of a new car, represented by \(y\), would also be expected to increase. This suggests a positive slope for the equation \(y=mx+b\).

Step 1: Understanding the Problem and Variables

Given the equation \(y=mx+b\) , where \(y\) is the average retail price of a new car, \(x\) is the number of years after 2000, \(m\) is the slope of the line and \(b\) is the y-intercept. It can be seen that the problem asks for the understanding of whether the slope \(m\) would be positive, negative or zero. This depends on the trend of car prices over time.

Step 2: Analyzing the Trend

Generally, the price of a new car will increase over time due to factors such as inflation, increased production costs, and advancements in technology. This would mean that as \(x\), years after 2000, increases, \(y\), the average retail price of a new car, would also increase, implying a positive relationship between \(x\) and \(y\).

Step 3: Determining the Slope

The slope \(m\) in the equation \(y=mx+b\) signifies the rate of change of \(y\) with respect to \(x\). Considering the previously determined trend of increasing car prices over time, this means the slope would be positive. As \(x\) increases, \(y\) also increases, implying a positive slope