Problem 28 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=e^{x}\) by basic transformations: (a) \(y=e^{-x}-1\) (b) \(y=-e^{x}+1\) (c) \(y=-e^{x-3}-2\)

Step-by-Step Solution

Answer

(a) Reflect across y-axis, shift down 1. (b) Reflect across x-axis, shift up 1. (c) Shift right 3, reflect across x-axis, shift down 2.

Step 1: Understanding the Base Function

The base function given is \(y = e^x\), which is an exponential function. It represents exponential growth, where the output increases exponentially as \(x\) increases.

Step 2: Transforming for (a)

For \(y = e^{-x} - 1\), first observe \(e^{-x}\). This represents a reflection of \(e^x\) across the \(y\)-axis because substituting \(-x\) inverts the function: \(e^{x} \rightarrow e^{-x}\). Next, the function \(e^{-x} - 1\) involves a downward shift by 1 unit, due to the \(-1\) outside the exponential function.

Step 3: Transforming for (b)

In \(y = -e^{x} + 1\), the negative sign in front of \(e^x\) reflects the graph across the \(x\)-axis: \(e^x \rightarrow -e^x\). Then, adding 1 to \(-e^x\) shifts the entire graph upward by 1 unit.

Step 4: Transforming for (c)

Consider \(y = -e^{x-3} - 2\). First, the term \(x-3\) means a horizontal shift to the right by 3 units: \(e^x \rightarrow e^{x-3}\). Secondly, the negative sign reflects the function across the \(x\)-axis: \(e^{x-3} \rightarrow -e^{x-3}\). Lastly, subtracting 2 translates the graph downward by 2 units.

Key Concepts

Function Transformations

Function transformations allow us to change the position or shape of a graph derived from the base function. Consider the base function, which in our exercise is the exponential function given by \( y = e^x \). Transformations can involve scaling, reflecting, or translating the graph in the plane.

Common types of transformations include:

• Reflection: Flipping the graph over a specific axis.
• Translation: Moving the graph horizontally or vertically.
• Stretching or Compressing: Changing the size and shape of the graph by scaling.

At the core of transforming a function like \( y = e^x \) is the understanding that these adjustments can express different real-world scenarios, such as different types of growth or decay by applying simple changes to the base function.

Reflection Across Axes

Reflection alters a graph by flipping it over a specified axis. This transformation changes the direction of growth or decay in the context of exponential functions. For our exercise, reflections are observed in the transformations of the exponential function:

### Reflecting Across the Y-Axis
When the input variable \( x \) is replaced with \( -x \), it reflects the graph across the y-axis. For example, \( y = e^{-x} \) is the mirror image of \( y = e^x \) over the y-axis. This reflection indicates a switch from exponential growth to decay, or vice versa.

### Reflecting Across the X-Axis
The addition of a negative sign in front of an exponential function reflects it across the x-axis. Thus, \( y = -e^x \) inverts \( y = e^x \), reflecting exponential values into negative values, which affects how outputs grow, converting growth into decline in this context.

Shifting Graphs

Shifting involves translating the graph either horizontally or vertically, which helps adjust the starting point of the function, allowing for a range of scenarios.

### Horizontal Shifts
Horizontal shifts occur when the x-variable is modified within the function. In the expression \( y = e^{x-3} \), shifting the graph to the right by 3 units moves every point on the graph 3 units to the right. Here, subtracting 3 inside the exponent indicates this rightward shift.

### Vertical Shifts
Vertical shifts happen when a constant is added or subtracted from the entire function. In \( y = e^{x} - 1 \), the \(-1\) subtracts 1 from every y-value on the graph, translating it downward by 1 unit. Conversely, \( y = -e^{x} + 1 \) showcases an upward shift by 1 unit due to the positive addition of a constant outside the function.

These shifts enable us to tailor the base function to meet specific situational requirements, showcasing an amazing flexibility within mathematical models.