Problem 33 - Algebra for College Students

Question

Graph each of the exponential functions. $$ f(x)=2^{x}-3 $$

Step-by-Step Solution

Answer

The graph of \( f(x) = 2^x - 3 \) is an exponential curve, shifted 3 units down, with a horizontal asymptote at \( y = -3 \).

Step 1: Understanding the Function

The function given is an exponential function of the form \( f(x) = 2^x - 3 \). This means it is a transformation of the base exponential function \( g(x) = 2^x \), translated 3 units downward.

Step 2: Determine Key Points

Identify key points of the basic function \( g(x) = 2^x \). Common points to plot are \((0, 1)\), \((1, 2)\), and \((-1, 0.5)\). For \( f(x) = 2^x - 3 \), subtract 3 from each y-value: \((0, -2)\), \((1, -1)\), and \((-1, -2.5)\).

Step 3: Plot the Key Points

Plot the points \((0, -2)\), \((1, -1)\), and \((-1, -2.5)\) on the coordinate plane. These are derived from shifting the corresponding points of the function \( g(x) = 2^x \) down by 3 units.

Step 4: Draw the Graph

Connect the plotted points with a smooth, continuous curve. The curve should approach the horizontal asymptote \( y = -3 \) as \( x \) continues to decrease, while increasing steeply as \( x \) increases.

Step 5: Identify the Asymptote

Since the function is of the form \( f(x) = 2^x - 3 \), recognize the horizontal asymptote is \( y = -3 \). The curve gets closer but never actually touches this line.

Key Concepts

Graphing Exponential Functions

Exponential functions are functions where a constant base is raised to a variable exponent, such as in our function, \( f(x) = 2^x - 3 \). When graphing these functions, it's essential to start by understanding the behavior of the base function. In this case, the base function \( g(x) = 2^x \) exhibits rapid growth. This growth is characteristic of exponential functions, which increase faster than linear or polynomial functions as the input \( x \) becomes larger.

When graphing any exponential function, it's helpful to first plot key points from the base function, and then consider any transformations such as shifts up, down, left, or right. In this exercise, the function includes a downward shift, altering the graph significantly from the base function.

Transformations of Functions

Transformations of functions involve altering the graph in certain predictable ways. In our specific function \( f(x) = 2^x - 3 \), there is a vertical shift. The transformation here is a downward shift by 3 units, as indicated by the "-3" outside the exponential expression.

Whenever you see "-c" in a function \( f(x) = a^x - c \), this indicates a downward vertical shift by \( c \) units. Conversely, a "+c" would indicate an upward shift. These vertical transformations affect the overall position of the graph on the y-axis, without changing its shape.

• Always be careful to adjust the y-values of key points when applying shifts.
• This understanding helps in correctly plotting the new graph positions relative to the original base function.

Horizontal Asymptotes

An essential aspect of understanding exponential functions is the concept of horizontal asymptotes. These are horizontal lines that the graph of the function will approach but never actually touch or cross.

For the function \( f(x) = 2^x - 3 \), the horizontal asymptote is \( y = -3 \). This occurs because the transformation shifts the base function \( 2^x \) down by 3 units. The closer \( x \) gets to negative infinity, the closer the function value approaches the asymptote, but it will never equal \( -3 \).

• Horizontal asymptotes help in predicting the end behavior of graphs.
• They provide a line of reference that indicates how low or high the exponential graph will extend.

Identifying Key Points

Identifying key points is crucial when graphing exponential functions. For \( f(x) = 2^x - 3 \), start with key points from the base function \( g(x) = 2^x \), such as \( (0, 1), \ (1, 2), \ and \ (-1, 0.5)\).

To account for the downward shift in \( f(x) \), subtract 3 from each y-coordinate:

• \((0, 1) \rightarrow (0, -2)\)
• \((1, 2) \rightarrow (1, -1)\)
• \((-1, 0.5) \rightarrow (-1, -2.5)\)

By plotting these transformed coordinates, you can draw the graph accurately.

This method allows for an intuitive understanding of how shifting affects location on the graph, helping solidify the general shape and direction of exponential functions.