Problem 121 - College Algebra

Question

Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: \(I(x)=1,054 x+23,286,\) where \(x\) is the number of years after 1990 . Which of the following interprets the slope in the context of the problem? a. As of 1990 , average annual income was \(\$ 23,286\). b. In the ten-year period from \(1990-1999\), average annual income increased by a total of \(\$ 1,054\). c. Each year in the decade of the \(1990 \mathrm{~s}\), average annual income increased by \(\$ 1,054\). d. Average annual income rose to a level of \(\$ 23,286\) by the end of \(1999 .\)

Step-by-Step Solution

Answer

The correct interpretation is c: Each year in the decade, income increased by $1,054.

Step 1: Identify the Linear Function

The given function is \( I(x) = 1,054x + 23,286 \). Here, \( x \) is the number of years after 1990.

Step 2: Understand the Components of the Function

A linear function is typically expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this function, \( m=1,054 \) and \( b=23,286 \).

Step 3: Interpret the Slope

The slope \( m = 1,054 \) represents the rate of change in average annual income per year. Therefore, it indicates how much the average annual income increases each year.

Step 4: Evaluate the Options

Examine each option: - a. Discusses the y-intercept, not the slope. - b. Incorrectly relates the slope to a total increase over ten years. - c. Correctly describes the slope as the yearly increase of income. - d. Misinterprets the values regarding the slope and year 1999.

Step 5: Select the Correct Interpretation

Option c accurately describes the slope's role in representing the yearly increase in average annual income.

Key Concepts

Slope Interpretation

The slope of a linear function is key to understanding how quantities change over time. In the given problem, the linear function is expressed as \( I(x) = 1,054x + 23,286 \).The slope here is \( 1,054 \), suggesting a specific change in average annual income each year. Slope is essentially the "steepness" of the line in our graph representation. It indicates how one variable changes in relation to another. In this context:

• \( m = 1,054 \) signifies the annual increase in income.
• Each year, the average income rises by \\(1,054.

Interpreting slope correctly is crucial as it gives a clear picture of the trend over time with respect to increase or decrease. Option c from the exercise correctly explains that, each year during the 1990s, the income increased by \\)1,054—which is a direct interpretation of the slope.

Linear Function Components

Understanding linear functions involves recognizing their components: the slope \( m \) and the y-intercept \( b \). A linear equation is often in the form of\( y = mx + b \).- **Slope (m):** Indicated by \( m = 1,054 \) in this function, it tells us how much the dependent variable (income) changes with each step in the independent variable (years).- **Y-intercept (b):** Given as \( b = 23,286 \), it represents the starting value or the point at which the line crosses the y-axis. This particular value shows the average income during the base year, which is 1990.These components offer clarity on how a linear function behaves. The slope offers insights into trends, while the y-intercept provides a baseline or starting value from which changes are measured.

Rate of Change

The rate of change in a linear function highlights how significantly one variable shifts in relation to another. This rate is represented by the slope in linear equations like\( I(x) = 1,054x + 23,286 \).- Here, the rate of change is \( 1,054 \), which describes how much the average annual income increases with each passing year.Key points to consider about the rate of change:

• It offers a constant rate at which one variable increases or decreases relative to another—to calculate, look at the change in the dependent variable divided by the change in the independent variable.
• In this problem, the rate of change helps predict future values, indicating a steady \$1,054 rise per year in income for the decade mentioned.

Recognizing the rate of change is helpful for understanding trends and making projections based on existing linear relationships.