Problem 34 - College Algebra 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

You would expect 'm' to be positive because car prices generally rise over time.

Step 1: Understanding the formula

In the formula \(y=mx+b\), 'm' denotes the slope of the line. The slope determines how \(x\) (the independent variable) affects \(y\) (the dependent variable). A positive slope indicates that as \(x\) increases, \(y\) also increases. A negative slope implies that as \(x\) increases, \(y\) decreases. A slope of zero means \(y\) is constant and does not change with \(x\).

Step 2: Applying the context

The formula in the problem statement is modeling the average retail price of a new car \(x\) years after 2000. This implies that \(x\) stands for the number of years after 2000, and \(y\) represents the average car price in those particular years. The slope 'm' in this context would represent the change in car prices per year.

Step 3: Interpreting the slope

To determine whether \(m\) is positive, negative, or zero, think about the typical trend in car prices over time. Generally, the cost of new cars tends to increase over time due to factors such as inflation, technological advancements, and higher manufacturing costs. Therefore, it could be expected that as the number of years past 2000 (x) increases, the average price of cars (y) would also increase.