Problem 44 - 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.) $$ \left(x_{1}, y_{1}\right)=(-1,4),\left(x_{2}, y_{2}\right)=(2,8) $$

Step-by-Step Solution

Answer

The relationship between \(x\) and \(y\) is \(y = 4\sqrt[3]{2} \cdot (\sqrt[3]{2})^x\).

Step 1: Understand the Problem

We're given two points and need to determine the relationship between variables when \(\log y\) is a function of \(x\). Since a straight line is formed when \(\log y\) is plotted against \(x\), this implies an exponential relationship between \(x\) and \(y\) in the form \(y = ab^x\).

Step 2: Use the Given Points

Use the points \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (2, 8)\). Substituting each point in the equation \(y = ab^x\) gives us two equations: 1. \(4 = ab^{-1}\),2. \(8 = ab^2\).

Step 3: Solve for a and b

From the first equation: \(ab^{-1} = 4\) \(\Rightarrow a = 4b\). Substitute this into the second equation: \(8 = 4bb^2\),leading to \(8 = 4b^3\).

Step 4: Calculate Value of b

Solve \(8 = 4b^3\) for \(b\):\[b^3 = 2\]\[b = 2^{1/3} = \sqrt[3]{2}.\]

Step 5: Substitute b and Solve for a

Substitute \(b = \sqrt[3]{2}\) back into \(a = 4b\):\[a = 4\sqrt[3]{2}.\]

Step 6: Express the Functional Relationship

Now that we have \(a\) and \(b\), the exponential relationship is \(y = 4\sqrt[3]{2} \cdot (\sqrt[3]{2})^x\).

Key Concepts

Understanding Exponential Relationships

In mathematics, exponential relationships describe a situation where one quantity grows or decays at a rate proportional to its current value. Think of it like interest on your savings account, where the interest amount depends on how much you currently have. This kind of relationship is denoted by the formula \( y = ab^x \), where:

• \( y \) is the dependent variable,
• \( x \) is the independent variable,
• \( a \) is a constant representing the initial value of \( y \) when \( x = 0 \),
• \( b \) is the base of the exponential function, indicating the growth factor.

When \( b > 1 \), \( y \) increases, showing growth. Conversely, when \( b < 1 \), \( y \) decreases, indicating decay.
For our exercise, substituting points into \( y = ab^x \) helps us find \( a \) and \( b \), revealing the specific growth pattern of the data.

Exploring Log-Linear Plots

A log-linear plot is a type of graph where one axis is on a logarithmic scale. In this problem,

• we graph \( \log y \) against \( x \).
• Logarithmic transformations often linearize exponential growth relationships.

By using a log-linear plot, the exponential relationship between \( x \) and \( y \) can be simplified into a linear form, turning complex equations into a visual straight line.
For instance, the function \( y = ab^x \) becomes a linear equation \( \log y = \log a + x \log b \). This transformation helps determine the base of exponentials (\( b \)) and initial value (\( a \)). It’s a handy technique for analyzing any dataset showing exponential trends and predicting future values.

Effective Graphing Techniques

Graphing serves as a powerful method to interpret and understand data, especially when assessing exponential relationships. Here are a few techniques to consider:

• Choose the Right Plot Type: In this context, using a log-linear plot makes the exponential relationship appear linear. Choose plots based on your data characteristics.
• Determine Key Points: Plot key points provided in the data set such as \((-1, 4)\) and \((2, 8)\). These guide the line of best fit.
• Use a Straight Edge: To accurately connect points, ensuring they form a straight line.

These simple techniques improve the accuracy and readability of your plots, making complex mathematical relationships easier to comprehend visually. This will be particularly useful if you are dealing with scientific or statistical data that follow an exponential trend.