91Ó°ÊÓ

Growth in height: Between the ages of 7 and 11 years, a certain boy grows 2 inches taller each year. At age 9 he is 48 inches tall. a. Explain why, during this period, the function giving the height of the boy in terms of his age is linear. Identify the slope of this function. b. Use a formula to express the height of the boy as a linear function of his age during this period. Be sure to identify what the letters that you use mean. c. What is the initial value of the function you found in part \(b\) ? d. Studying a graph of the boy's height as a function of his age from birth to age 7 reveals that the graph is increasing and concave down. Does this indicate that his actual height (or length) at birth was larger or smaller than your answer to part c? Be sure to explain your reasoning.

Step by step solution

01

Understanding Linear Growth

The boy grows by a constant amount, 2 inches, every year between the ages of 7 and 11. This consistent rate of growth indicates a linear relationship between age and height. Linear functions have the form \( y = mx + b \), where \( m \) is the slope of the function. The slope, \( m \), is the rate of change, which in this case is 2 inches per year.

02

Deriving the Linear Equation

We know from the information given that at age 9, the boy is 48 inches tall. We can express the height \( h \) of the boy as a function of his age \( a \) in years using the linear function format: \( h = m(a - 9) + 48 \). Here, \( m = 2 \), so the formula becomes \( h = 2(a - 9) + 48 \), which simplifies to \( h = 2a - 18 + 48 \) and further to \( h = 2a + 30 \). Here, \( a \) is the age of the boy, and \( h \) is his height in inches.

03

Identifying Initial Value

The initial value of a linear function is the value of the function when the independent variable is zero. However, for this problem, initial value typically refers to the boy's height at a meaningful starting age within the discussed range. By plugging \( a = 7 \) into \( h = 2a + 30 \), we find \( h = 2(7) + 30 = 44 \) inches, which closely reflects his height when this linear growth started.

04

Analyzing Graph and Initial Height at Birth

A graph that is increasing and concave down indicates that although the height (or length) of the boy at birth was increasing, the rate of growth was decreasing over time. This suggests that his height at birth was likely smaller than the initial value at age 7 found in part \( b \), which was 44 inches, because concave down indicates an initially slower rate of growth that accelerates later.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate of Change

In the context of linear functions, the "rate of change" refers to how much a dependent variable changes in response to a change in the independent variable. For the boy's growth scenario, this is represented by how much taller he becomes with each passing year. Here, the rate of change is constant at 2 inches per year.
This consistent increase ensures that as the boy ages from 7 to 11, his height follows a predictable pattern, making the relationship between age and height linear.

• The rate of change is also called the slope in the equation of a linear function.
• In mathematical terms, if the function is expressed as \( y = mx + b \), then \( m \) represents the rate of change.
• A constant rate of 2 inches means a steady and uniform growth over each year.

The idea of "rate of change" is crucial for understanding linear relationships, providing insight into how variables interact in a straightforward, predictable manner.

Initial Value

The initial value in a linear function represents the starting point or the value of the dependent variable when the independent variable is zero. In practical applications, it often aligns with a significant or starting point within a given range.
For the boy's height equation, the function is written as \( h = 2a + 30 \). Here, 30 is not the initial value for age 0, but it helps determine his height at any age within the given period.

• To find the height when linear growth begins, use the equation for age 7: \( h = 2(7) + 30 = 44 \) inches.
• This value of 44 inches represents the height from which linear growth starts, rather than a true initial at age 0.
• 
  Initial values in linear functions provide a foundation for determining other values, establishing a baseline for calculations.

Understanding the initial value adds clarity, enabling interpretations of linear slope equations, especially when applied to real-world scenarios like growth patterns.

Graph Analysis

Graph analysis is an essential skill in understanding linear functions. By plotting a function on a graph, you can visually ascertain how the dependent variable (boy's height) changes with respect to the independent variable (age).
In our context, the graph of the boy's height from ages 7 to 11 would depict a straight line, indicating consistent growth with a steady rate of change.

• This straight line would have a positive slope, signifying an increase of 2 inches per year.
• The y-intercept of the line can provide the initial value or the height from which the boy began his linear growth pattern.
• When assessing the graph from birth to age 7, a curve that is increasing and concave down suggests slower initial growth, followed by acceleration.

Utilizing graph analysis allows students to connect algebraic concepts to visual representations, enhancing comprehension of linear and non-linear growth trends across time.

Linear Growth

Linear growth describes a process of consistent, steady increase. It signifies equal increments over equal intervals of the independent variable. For the boy growing in height, linear growth means a uniform addition of 2 inches every year between ages 7 and 11.

• Linear growth can be represented by a straight line on a graph, with the slope indicating the rate of change.
• It's defined by the equation of the form \( y = mx + b \), where changes are orderly and predictable.
• In contrast, non-linear growth would involve fluctuating rates of change, resulting in curves rather than straight lines.
• Linear models are valuable for forecasting and planning, as they imply stability and predictability over time.

Understanding linear growth provides a robust framework for analyzing and predicting patterns in various fields, whether it be biology or economics, fostering better-informed decisions and insights.