Question

(a) Sketch a graph of the exponential function \(y=A e^{r t} ;\) indicate the value of the vertical intercept. (b) Then sketch the graph of the log function \(t=\frac{\ln y-\ln A}{r},\) and indicate the value of the horizontal intercept.

Short answer

The vertical intercept of the exponential function is \( (0, A) \). The horizontal intercept of the log function is \( y = A \).

Step-by-step solution

Understand the Exponential Function

The exponential function is given by the equation \( y = A e^{rt} \). This function represents exponential growth or decay depending on the value of \( r \). \( A \) is the initial value (the vertical intercept).

Determine the Vertical Intercept of the Exponential Function

To find the vertical intercept of the function \( y = A e^{rt} \), we set \( t = 0 \). This gives \( y = A e^{r \times 0} = A e^{0} = A \). Therefore, the vertical intercept is \( (0, A) \).

Sketch the Exponential Function Graph

When sketching the graph, note that if \( r > 0 \), the graph increases as \( t \) increases, showing exponential growth. If \( r < 0 \), the graph decreases, showing exponential decay. It always passes through the point \( (0, A) \).

Understand the Logarithmic Function

The logarithmic function is given by \( t = \frac{\ln y - \ln A}{r} \). This can be rewritten as \( t = \frac{1}{r} (\ln y - \ln A) \). This is a linear function in terms of \( \ln y \).

Find the Horizontal Intercept of the Logarithmic Function

To find the horizontal intercept, set \( t = 0 \). Solve for \( \ln y \): \( 0 = \frac{\ln y - \ln A}{r} \) implies \( \ln y - \ln A = 0 \) or \( \ln y = \ln A \). Therefore, \( y = A \) when \( t = 0 \).

Sketch the Logarithmic Function Graph

The graph of \( t = \frac{\ln y - \ln A}{r} \) is a straight line with a slope of \( \frac{1}{r} \) and passes through the point where \( y = A \), which is the horizontal intercept when \( y = A \). If \( r > 0 \), the slope is positive, and if \( r < 0 \), the slope is negative.

Key concepts

Logarithmic Functions

Logarithmic functions are closely related to exponential functions. They are essentially the inverse, which means that a logarithmic function can "undo" what an exponential function does. In the context of the given problem, the logarithmic function is \( t = \frac{\ln y - \ln A}{r} \). This transformation involves the natural logarithm, \( \ln \), which is the logarithm to the base \( e \).

• \( \ln y \) represents the natural log of the variable \( y \).
• \( \ln A \) is a constant representing the natural log of the initial value \( A \).

The equation can be rewritten as a linear equation in terms of \( \ln y \):

\[ t = \frac{1}{r} (\ln y - \ln A) \]

Key Features of Logarithmic Functions

• Logarithmic functions express the relationship of how one variable changes in a multiplicative manner with respect to another.
• The graph of a logarithmic function is a straight line when plotted with respect to \( \ln y \).
• The slope of this line is influenced by \( \frac{1}{r} \), making the line steeper if \( |r| \) is small, and less steep if \( |r| \) is large.

Graph Sketching

Graph sketching is a useful skill in understanding the behavior of functions. It involves drawing a rough plot of a function to give insight into its shape and key characteristics.

In this exercise, you are required to sketch both an exponential and a logarithmic function.

Steps for Exponential Function

• An exponential function like \( y = A e^{rt} \) will show growth if \( r > 0 \) and decay if \( r < 0 \).
• The graph will always pass through \( (0, A) \), the vertical intercept.
• As \( t \) approaches infinity, the function rises (if \( r > 0 \)) or approaches zero (if \( r < 0 \)).

Steps for Logarithmic Function

• For the logarithmic function \( t = \frac{\ln y - \ln A}{r} \), sketch it as a straight line when plotting \( t \) versus \( \ln y \).
• The slope is \( \frac{1}{r} \), influencing the steepness of the line.
• Identify the horizontal intercept at \( y = A \) to guide your drawing.

Intercepts

Intercepts are the points where a graph crosses the axes. They provide a valuable way to understand key points of a function easily.

Vertical Intercept in Exponential Functions

• The vertical intercept is where the function crosses the y-axis, at \( t = 0 \).
• For the exponential function \( y = A e^{rt} \), it simplifies to \( y = A \) when \( t = 0 \), meaning the vertical intercept is \( (0, A) \).

Horizontal Intercept in Logarithmic Functions

• The horizontal intercept is where the function crosses the x-axis, specifically where \( t = 0 \).
• For the logarithmic function, you set \( t = 0 \) and solve for \( y \), ultimately finding \( y = A \).
• This means that the horizontal intercept occurs at \( y = A \).

Understanding intercepts is crucial as they indicate fundamental properties such as the initial value in an exponential growth context or the point of equality in a logarithmic relationship.