Problem 31 - College Algebra

Question

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=-2^{x}\)

Step-by-Step Solution

Answer

The function \(g(x)=-2^{x}\) is a vertical reflection of \(f(x)=2^{x}\). When graphed, it creates a downward-sloping curve that crosses the y-axis at -1.

Step 1: Graph the original function

Start by graphing the function \(f(x) = 2^x\). This is an exponential function, and its graph should look like an upward-sloping curve that crosses the y-axis at \(f(0) = 2^0 = 1\). The graph becomes steeper to the right of the y-axis (for \(x > 0\)) and approaches the x-axis but never crosses it to the left of the y-axis (for \(x < 0\)).

Step 2: Understand the transformation

The function \(g(x) = -2^x\) is a vertical reflection of \(f(x) = 2^x\). This means that all y-values will be the opposite, or negative, of what they were in the original function to turn the graph upside down.

Step 3: Graph the transformed function

Graphically, this transformation results in a downward sloping curve which crosses the y-axis at \(-2^0 = -1\). The graph becomes steeper to the left of the y-axis (for \(x < 0\)) and it approaches the x-axis but never crosses it to the right of the y-axis (for \(x > 0\)).

Step 4: Confirm the graph with a graphing utility

Now use a graphing tool (like a graphing calculator or an online graphing utility) to confirm that your hand-drawn graph of \(g(x) = -2^x\) is accurate.

Key Concepts

Exponential Function

Understanding an exponential function is essential when dealing with a variety of mathematical and real-world applications. This type of function is generally expressed as \( f(x) = a^x \), where \( a \) is a constant and \( x \) is the exponent. One of the distinguishing characteristics of an exponential function is that the rate of change increases or decreases exponentially, making the function's growth or decay very rapid.

As we visualize exponential growth through the example \( f(x) = 2^x \), we should recognize that the base, in this case 2, is greater than 1, so the function exhibits growth as \( x \) increases. The graph crosses the y-axis at \( f(0) = 1 \), since any nonzero number raised to the power of 0 equals 1. To the right of the y-axis, the graph rapidly ascends, and to the left, it gently approaches the x-axis, yet it never touches it, showing that the function has an asymptote along the x-axis.

Vertical Reflection

A vertical reflection of a function flips the graph over the x-axis. Mathematically, if you have a function \( f(x) \), its vertical reflection is \( g(x) = -f(x) \). This essentially multiplies every y-value of the original function by -1, changing the sign of the output values, which inverts the graph.

When we apply a vertical reflection to the exponential function \( f(x) = 2^x \), we obtain \( g(x) = -2^x \). This negates every y-value of the original function, so a point that was at (2, 4) on \( f(x) \) will be at (2, -4) on \( g(x) \), for example. Moreover, the behavior of the function changes: where the original graph increased, the reflected graph decreases and vice versa. The graph of \( g(x) = -2^x \) will descend to the right and approach the x-axis without crossing it, creating a reflection image of the \( f(x) \) function.

Graph Transformations

Graph transformations involve making changes to the original graph of a function, such as moving it up or down, left or right, stretching it, compressing it, or reflecting it across an axis. These transformations can be represented by modifications to the function's formula.

In our example, transforming \( f(x) = 2^x \) into \( g(x) = -2^x \) showcases a vertical reflection, which is a specific type of transformation. It's important to note that transformations follow specific rules. For instance, adding or subtracting a constant from the function shifts the graph vertically, multiplying the function's output by a constant results in vertical stretching or compressing, and multiplying the input variable \( x \) by a constant leads to horizontal stretching or compressing.

Coordinate Plotting

To properly graph functions, plotting coordinates of key points is an essential skill. By choosing input values for \( x \) and calculating the corresponding output, \( y \( or \) f(x) \), we obtain points in the form \( (x, y) \) that can be plotted on a graph.

For exponential functions like \( f(x) = 2^x \) and its reflection \( g(x) = -2^x \), it's useful to select a range of positive and negative integer values for \( x \) to see how rapidly the function grows or declines. Plotting these coordinates provides a visual representation of the graph and can help in understanding the overall behavior of the function. In practice, after plotting key points, you would sketch the curve using those points as a guide to ensure accuracy.