Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=e^x, g(x)=e^{2 x}\)

Short answer

Function \( g(x) = e^{2x} \) is the result of taking the graph of the base function \( f(x) = e^x \) and stretching it horizontally by a factor of 1/2.

Step-by-step solution

Establish the Base Function

Identify the base function that we are starting with, which is \( f(x)=e^x \). This is an exponential function where the base is the mathematical constant \( e \). The graph of \( f \) is upward-sloping, and crosses the y-axis at \( (0,1) \).

Understand the Transformation

Recognize the change made to the base function to create \( g(x) = e^{2x} \). Here, the input \( x \) is being multiplied by 2. This stretches the graph of \( f \) horizontally by a factor of 1/2.

Graphing the Functions

Use a graphing tool to plot both \( f(x) = e^x \) and \( g(x) = e^{2x} \). Both graphs will cross the y-axis at (0,1), but \( g(x) = e^{2x} \) will appear narrower than \( f(x) = e^x \) due to the horizontal stretch.

Key concepts

Exponential Function

An exponential function is a type of mathematical function that rapidly increases or decreases as the input grows. It is generally expressed in the form \( f(x) = a^{x} \), where \( a \) is a positive constant. In this case, our exponential function is \( f(x) = e^x \), where \( e \) is the mathematical constant approximately equal to 2.718. Exponential functions have unique characteristics:

• They have a constant base.
• They are defined for all real numbers.
• The range is always positive real numbers.
• They exhibit continuous and smooth curves.

For \( f(x) = e^x \), the graph shows a continuous increase, crossing the y-axis at \( (0,1) \). This point is known as the y-intercept, and occurs here because \( e^0 = 1 \). Understanding these foundational properties is essential when manipulating exponential functions.

Horizontal Stretch

In mathematics, transformations can modify the size and shape of a graph. A horizontal stretch is one such transformation that alters the width of a graph along the x-axis. When we multiply the input \( x \) by a constant, the graph stretches or compresses horizontally.For the function \( g(x) = e^{2x} \), this horizontal transformation is performed by multiplying the \( x \) variable by 2. It may seem counterintuitive, but this actually compresses the graph horizontally by a factor of \( \frac{1}{2} \). So if you were to compare \( f(x) = e^x \) with \( g(x) = e^{2x} \), the graph of \( g(x) \) will appear narrower.Key points to remember for horizontal stretches:

• If \( x \) is multiplied by a value greater than 1, the graph is compressed (narrower).
• If \( x \) is multiplied by a value between 0 and 1, the graph is stretched (wider).

Graphing Functions

Graphing functions is a crucial skill in math, as it allows us to visualize and understand the behavior of various functions, including transformations. For exponential functions like \( f(x) = e^x \) and its transformed version \( g(x) = e^{2x} \), graphing reveals important differences and similarities.Both functions cross the y-axis at the same point \((0,1)\) because \( e^0 = 1 \), regardless of the stretch factor. However, their shapes indicate the effect of the transformation. The original function, \( f(x) = e^x \), has a wider, more gradual slope. In contrast, enhanced by a horizontal compression, \( g(x) = e^{2x} \) rises or falls more steeply given the same change in \( x \).To effectively graph these functions:

• Use graphing tools or software for accuracy.
• Identify key points like the y-intercept first.
• Note changes such as narrower shapes or steeper inclines.

Graphing provides a visual context that builds deeper understanding of how algebraic transformations like horizontal stretches impact function behavior.