Question

Without using a graphing utility, sketch the graph of \(y=2^{x} .\) Then, on the same set of axes, sketch the graphs of \(y=2^{-x}, y=2^{x-1}, y=2^{x}+1,\) and \(y=2^{2 x}\).

Short answer

Question: Sketch the following exponential functions on the same set of axes and describe the transformations applied to each function: \(y = 2^x\), \(y = 2^{-x}\), \(y = 2^{x-1}\), \(y = 2^x + 1\), and \(y = 2^{2x}\). Answer: The functions represent the basic exponential function \(y = 2^x\), its reflection across the y-axis (\(y = 2^{-x}\)), a horizontal shift to the right (\(y = 2^{x-1}\)), a vertical shift up (\(y = 2^x + 1\)), and a vertical stretch (\(y = 2^{2x}\)). Sketch these functions on the same set of axes, showing the transformations applied to the basic exponential function to obtain each graph.

Step-by-step solution

Understand the basic exponential function

The basic exponential function has the form \(y = a^{x}\), where a is a positive constant. In this case, we have \(y = 2^x\). The graph of this function increases rapidly as x increases, and approaches zero as x decreases. The graph passes through the point (0, 1) because any number to the power of 0 equals 1.

Reflect across the y-axis

The function \(y = 2^{-x}\) is the reflection of \(y = 2^x\) across the y-axis. To see this reflection, note that \(2^{-x} = (\frac{1}{2})^x\). So, we replace every x in the original function with -x. This transformation "flips" the graph.

Translate to the right

The function \(y = 2^{x-1}\) is the transformation of \(y = 2^x\) shifted one unit to the right. We obtain this function by replacing every x in the original function with \((x-1)\). The graph shifts to the right and still increases rapidly as x increases and approaches zero as x decreases.

Translate up

The function \(y = 2^x + 1\) is the transformation of \(y = 2^x\) shifted one unit up. We obtain this function by adding 1 to the original function. The graph still increases rapidly as x increases and now approaches 1 as x decreases.

Stretch vertically

The function \(y = 2^{2x}\) is the transformation of \(y = 2^x\) with a vertical stretch. We obtain this function by replacing every x in the original function with \(2x\). The graph still increases rapidly as x increases and approaches zero as x decreases, but the rate of increase is faster. Now, combine all these graphs on the same set of axes. Since we cannot draw the graphs here, remember to apply the given transformations one at a time to obtain the desired sketch for each function.

Key concepts

Graph Transformations

Graph transformations can be seen as modifications to the original graph, changing its appearance while maintaining its fundamental shape. When dealing with exponential functions, these transformations include various alterations such as reflections, translations, and stretches. The original function considered here is the basic exponential function, \( y = 2^x \). This function forms the basis for all subsequent transformations.

With graph transformations, it's important to remember that:

• Alterations can change the position and orientation of the graph on the plane.
• Keeping track of the transformations applied can help you visualize the graph mentally or sketch it more accurately.

The key is to modify specific elements of the function and observe how these modifications impact the graph. By observing each transformation, we can gain deeper insights into the behavior of exponential functions.

Reflection Across Axes

Reflection across an axis involves flipping the graph over a line, creating a mirror image. When reflecting an exponential function across the y-axis, we replace \( x \) with \( -x \). For the function \( y = 2^x \), this reflection results in the function \( y = 2^{-x} \), or equivalently \( y = (\frac{1}{2})^x \). This transformation reverses the direction in which the function increases.

Here's what happens during a reflection across the y-axis:

• The graph is flipped horizontally, changing the direction of growth.
• Original points on the right of the y-axis move to the left and vice versa.

This alteration modifies how the exponential function behaves as x becomes negative and is a powerful tool to understand different orientations of graphs.

Translation of Functions

Translation of functions involves moving the graph either horizontally or vertically without altering its shape.

*Horizontal Translation:* For the exponential function \( y = 2^x \), translating one unit to the right results in \( y = 2^{x-1} \). This translation is achieved by replacing \( x \) with \( x-1 \).

• This shift moves the entire graph one unit towards positive x values.
• Despite the shift, the rate of increase remains the same.

*Vertical Translation:* Adding a constant to \( y = 2^x \), as in \( y = 2^x + 1 \), achieves a vertical translation. Here, the graph moves up one unit along the y-axis.

• This results in a new horizontal asymptote at y = 1.
• The rapid increase as x becomes positive continues unchanged.

Translations are useful for altering the position of graphs on a coordinate plane while maintaining their general form.

Vertical Stretch

A vertical stretch in a function occurs when the output value for each x is multiplied by a factor, making the graph narrower and steeper. For exponential functions, a vertical stretch could be applied by modifying the exponent itself.

In \( y = 2^{2x} \), replacing \( x \) with \( 2x \), effectively stretches the graph vertically,

• This results in a sharper increase in the graph as x increases.
• The shape gets steeper compared to the basic \( y = 2^x \) function.

The vertical stretch emphasizes the sensitivity of exponential growth, showing how quickly values can change with relatively small changes in x. Understanding vertical stretch helps in grasping the impact of exponential growth at varying rates.