Problem 20 - Calculus

Question

Find all the tables that have the given characteristic. (A) $$\begin{array}{c|c|c|c|c} \hline x & 0 & 40 & 80 & 160 \\ \hline y & 2.2 & 2.2 & 2.2 & 2.2 \\ \hline \end{array}$$ (B) $$\begin{array}{c|c|c|c|c} \hline x & -8 & -4 & 0 & 8 \\ \hline y & 51 & 62 & 73 & 95 \\ \hline \end{array}$$ (C) $$\begin{array}{c|c|c|c|c} \hline x & -4 & -3 & 4 & 6 \\ \hline y & 18 & 0 & 4.5 & -2.25 \\ \hline \end{array}$$ (D) $$\begin{array}{c|c|c|c|c} \hline x & 3 & 4 & 5 & 6 \\ \hline y & 18 & 9 & 4.5 & 2.25 \\ \hline \end{array}$$ \(y\) could be a linear function of \(x\)

Step-by-Step Solution

Answer

Tables (A) and (D) have linearly related values of \( y \).

Step 1: Understanding Linear Functions

A linear function can be expressed in the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. If the change in \( y \) is consistent for each unit change in \( x \), then the function is linear.

Step 2: Check Table (A)

In table (A), the \( y \) values are 2.2 for each \( x \). Here, \( y \ = 2.2 \) is constant and can be expressed as \( y = 0x + 2.2 \), which is a linear function where the slope \( m = 0 \). Thus, table (A) represents a linear function.

Step 3: Check Table (B)

Calculate the differences between consecutive \( y \) values: \( 62 - 51 = 11 \), \( 73 - 62 = 11 \), and \( 95 - 73 = 22 \). Since the changes in \( y \) are inconsistent, \( y \) is not a linear function of \( x \).

Step 4: Check Table (C)

Calculate the differences between consecutive \( y \) values: \( 0 - 18 = -18 \), \( 4.5 - 0 = 4.5 \), and \( -2.25 - 4.5 = -6.75 \). Because the change in \( y \) is not consistent, \( y \) is not a linear function of \( x \).

Step 5: Check Table (D)

Calculate the differences between consecutive \( y \) values: \( 9 - 18 = -9 \), \( 4.5 - 9 = -4.5 \), and \( 2.25 - 4.5 = -2.25 \). Since the ratio of \( rac{-9}{1} : rac{-4.5}{1} : rac{-2.25}{1} = 2:1 \) is consistent, it means \( y \) changes proportionally with \( x \), confirming \( y \) as a linear function.

Key Concepts

Slope

The slope is a key feature in understanding linear functions. It measures how steep the line is and determines the direction it moves - either upwards, downwards, or stays flat. In mathematical terms, slope is represented by the letter \( m \) in the linear equation \( y = mx + b \). This slope \( m \) is calculated as the ratio of the change in \( y \) to the change in \( x \), also known as "rise over run." This means that for each unit change in \( x \), the \( y \) value changes by \( m \) units.

• If \( m > 0 \), the line rises, moving up as \( x \) increases.
• If \( m < 0 \), the line falls, moving down as \( x \) increases.
• If \( m = 0 \), the line is horizontal, showing no rise or fall - like in Table (A) where the slope is 0.

Understanding slope is crucial because it shapes the function's behavior, indicating whether the relationship is increasing, decreasing, or constant.

Y-intercept

The y-intercept is another important element of linear functions. It tells you the point where the line crosses the y-axis. In the linear equation \( y = mx + b \), this is represented by \( b \). Essentially, it shows the value of \( y \) when \( x = 0 \). When examining tables such as (A) through (D), the y-intercept can sometimes be clearly seen. For Table (A), where \( y \) is constantly 2.2, the y-intercept \( b \) is 2.2. However, for other tables, you might need to calculate or visualize the line to identify \( b \).

• A positive y-intercept places the line above the origin on the graph.
• A negative y-intercept places the line below the origin.
• A y-intercept of zero means the line passes through the origin.

Understanding the y-intercept helps in graphing the line and predicting values of \( y \) for given values of \( x \).

Consistent Change

A consistent change in a function indicates linearity. For a function to be linear, the change in \( y \) must be consistent for each unit increase in \( x \). In simpler terms, the difference in \( y \) between consecutive points should be the same each time. This steady change is what makes the graph of a function a straight line. For example, in Step 2, Table (A) shows \( y \) values that are constant, demonstrating a consistent change of 0. This is still considered linear with a slope of 0. But for Table (D), the differences in \( y \) are \(-9\), \(-4.5\), and \(-2.25\), which consistently increase by the same proportion, making it linear.

• A constant increase indicates a positive slope, resulting in an upward line.
• A constant decrease indicates a negative slope, resulting in a downward line.
• No change indicates a slope of zero, as seen in horizontal lines.

Recognizing consistent changes helps you quickly identify linear relationships, crucial for solving problems involving functions.