Problem 13 - Essential Calculus Early Transcendentals

Question

Starting with the graph of \(y=e^{x},\) write the equation of the graph that results from (a) shifting 2 units downward (b) shifting 2 units to the right (c) reflecting about the \(x\) -axis (d) reflecting about the \(y\) -axis (e) reflecting about the \(x\) -axis and then about the \(y\) -axis

Step-by-Step Solution

Answer

(a) \(y = e^x - 2\), (b) \(y = e^{x-2}\), (c) \(y = -e^x\), (d) \(y = e^{-x}\), (e) \(y = -e^{-x}\).

Step 1: Understand the Original Function

The original function given is the exponential function \(y = e^x\). This is the base function which we will transform according to the given instructions.

Step 2: Shift Downward by 2 Units

To shift the graph of a function downward by 2 units, subtract 2 from the function's equation. Therefore, the equation after shifting downward is \(y = e^x - 2\).

Step 3: Shift to the Right by 2 Units

To shift the graph of a function 2 units to the right, replace \(x\) with \(x-2\) in the function's equation. The new equation is \(y = e^{x-2}\).

Step 4: Reflect About the X-axis

To reflect the graph across the x-axis, multiply the entire function by \(-1\). The transformed equation becomes \(y = -e^x\).

Step 5: Reflect About the Y-axis

To reflect the graph across the y-axis, replace \(x\) with \(-x\) in the function's equation. The modified equation is \(y = e^{-x}\).

Step 6: Reflect About the X and Y-axes

To reflect about both the x-axis and the y-axis, first reflect about the x-axis and then the y-axis. Apply both transformations to get \(y = -e^{-x}\).

Key Concepts

Graph Transformations

When dealing with graph transformations, we are essentially changing the way the graph of a function looks without altering its basic properties or exponential nature. Graph transformations include various operations that reposition and reshapes the graph in a coordinate plane.

Here are some ways you can transform graphs:

• Vertical Shifts: Moving the graph up or down by adding or subtracting a constant to the function.
• Horizontal Shifts: Moving the graph left or right by adding or subtracting inside the function's argument.
• Reflections: Flipping the graph over a specific line, such as the x-axis or y-axis.
• Stretches and Compressions: Altering the graph’s steepness by multiplying the function by a constant.

Learning these transformations allows you to predict and understand how a function's graph will look after any changes have been applied.

Function Reflection

Function reflection is a type of transformation that involves flipping the graph of a function over a specific axis. This transformation can be visualized as taking the graph and creating a mirror image of it.

For exponential functions, the process is simple:

• Reflection Over the X-axis: To reflect a function over the x-axis, negate the entire function, transforming \( y = e^x \) to \( y = -e^x \).
• Reflection Over the Y-axis: To reflect over the y-axis, substitute \( x \) with \( -x \), modifying \( y = e^x \) to \( y = e^{-x} \).

These reflections change the sign but not the essence of the exponential function, affecting how the function grows or decays across the axes.

Function Shifting

Shifting functions is another common transformation method in graphing, moving the graph around the plane without changing its shape. Shifts can be vertical or horizontal:

• Vertical Shifts: Achieved by adding or subtracting a constant from the function. For example, the graph of \( y = e^x \) shifts downward by 2 units when the equation becomes \( y = e^x - 2 \).
• Horizontal Shifts: Executed by altering the input \( x \) of the function. Shifting the function to the right involves replacing \( x \) with \( x-2 \), resulting in \( y = e^{x-2} \).

Function shifting is crucial for aligning graphs as desired without affecting their shape or inherent characteristics.

Exponential Equations

Exponential equations are expressions where variables appear in the exponent. They are a staple in mathematics, modeling situations where growth or decay is proportional to current value.

The simplest form, \( y = e^x \), is exponentially increasing since the base \( e \) is greater than 1. Graph transformations like shifts and reflections affect exponential equations by changing their position or orientation without altering their exponential nature.

Exponential equations:

• Model Real-World Phenomena: Such as population growth, radioactive decay, and interest calculations.
• Have Different Forms: By applying transformations such as reflections (changing the sign of terms) and shifts (modifying constants), the equation adapts to diverse scenarios.

Understanding these equations and applying transformations help in solving a variety of practical and theoretical problems.