Problem 45 - Calculus for Biology and Medicine

Question

When \(\log y\) is graphed as a function of \(x\), a straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional rela\mathrm{\\{} t i o n s h i p . ~ ( T h e ~ o r i g i n a l ~ \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-1,1),\left(x_{2}, y_{2}\right)=(1,3) $$

Step-by-Step Solution

Answer

The functional relationship is \(y = 1.729 \cdot 1.725^x\).

Step 1: Understand the problem

We need to find the functional relationship between variables \(x\) and \(y\) given that \(\log y\) versus \(x\) forms a straight line on a graph. The points provided are \((-1, 1)\) and \((1, 3)\). This indicates that the relationship is in the form of \(y \) as a power function of \(x\), such as \(y = ab^x\).

Step 2: Convert the y-values to logarithmic form

To find the linear relationship when \(\log y\) is graphed against \(x\), find the natural logarithms of the \(y\) values: \[ \log(1) = 0, \quad \log(3) = \log 3 \approx 0.4771 \]

Step 3: Form the linear equation

Since \(\log y\) vs \(x\) is a straight line, use the formula for a line, \(y = mx + c\). Here, \(\log y = mx + c\). Use the points \((-1,0)\) and \((1,0.4771)\) to calculate the slope: \[ m = \frac{0.4771 - 0}{1 - (-1)} = \frac{0.4771}{2} \approx 0.2385 \]

Step 4: Calculate the intercept

With the slope \(m = 0.2385\), use one of the points to find the intercept \(c\). Using point \((1, 0.4771)\), substitute in the line equation: \[ 0.4771 = 0.2385 \times 1 + c \] Solving gives \(c = 0.4771 - 0.2385 = 0.2386\). Thus, \(\log y = 0.2385x + 0.2386\).

Step 5: Determine the functional form of y

Transform \(\log y = 0.2385x + 0.2386\) back to exponential form: \[ y = 10^{(0.2385x + 0.2386)} \] This implies: \[ y = 10^{0.2386} \cdot 10^{0.2385x} = a \cdot b^x \] where \(a = 10^{0.2386}\) and \(b = 10^{0.2385}\). Calculate these values for the constants: \[ a \approx 10^{0.2386} \approx 1.729 \] \[ b \approx 10^{0.2385} \approx 1.725 \] So, the functional form is \(y = 1.729 \cdot 1.725^x\).

Key Concepts

Linear Relationships

A linear relationship in mathematics is one where two variables are related in such a way that a change in one variable causes a proportional and consistent change in the other. This can be visualized as a straight line on a graph.
When dealing with logarithmic transformations, like in this exercise where \( \log y \) is graphed against \( x \), this linear relationship implies a unique connection between these transformed variables. The general formula for such a linear relationship is \( y = mx + c \), where:

• \( y \) represents the dependent variable (here, \( \log y \))
• \( m \) is the slope of the line
• \( x \) is the independent variable
• \( c \) is the y-intercept where the line crosses the y-axis

In this log-linear plot, recognizing the linear relationship helps to uncover the exponential form of the original variables.

Exponential Form

The exponential form is a fundamental representation that helps describe growth or decay processes, like population growth or radioactive decay. It has the form \( y = ab^x \), where:

• \( a \) is the initial amount or y-intercept (when \( x = 0 \))
• \( b \) is the base of the exponential which determines the rate of growth or decay
• \( x \) is the exponent, often representing time or another independent variable

In the problem, by transforming the linear equation \( \log y = 0.2385x + 0.2386 \) back into exponential form, we deduced \( y = 10^{0.2386} \cdot 10^{0.2385x} \). This exponential function illustrates how \( y \) changes exponentially with \( x \), depicting a curve on the original x-y plot. Calculating these, we find \( a \approx 1.729 \) and \( b \approx 1.725 \), showing how \( y \) grows as \( x \) increases.

Slope and Intercept

Understanding the slope and intercept is crucial in analyzing graphs and functions. The slope \( m \) indicates how steep a line is and which direction it goes. If \( m > 0 \), the line ascends as you move along the x-axis; if \( m < 0 \), it descends.
In the context of this exercise, the slope of the line is \( m = 0.2385 \). This means for every one-unit increase in \( x \), \( \log y \) increases by \( 0.2385 \). This positive slope confirms that as \( x \) increases, \( y \) also increases exponentially.
The intercept \( c \) shows where the line crosses the y-axis. In this scenario, the intercept is \( c = 0.2386 \), meaning when \( x = 0 \), \( \log y = 0.2386 \). It helps determine the starting value of \( y \) when \( x \) is zero, as seen when converted from log to actual scale by \( y = 10^{0.2386} \). This results in an initial value of approximately \( y = 1.729 \). The slope and intercept thus are instrumental in reconstructing the functional relationships.