Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=3^x, g(x)=3^x+5\)

Short answer

The function \(g(x) = 3^x + 5\) is a transformation of the function \(f(x) = 3^x\), where the '+5' represents a vertical shift 5 units up. When graphing, \(g(x)\) would have an identical shape to \(f(x)\), just shifted up 5 units on the y-axis.

Step-by-step solution

Identifying the transform

Here, the given function \(f(x) = 3^x\) is transformed into \(g(x) = 3^x + 5\). The '+5' in \(g(x)\) represents a vertical shift of the function \(f(x)\) in the positive y-direction, meaning the graph of the function will move upwards by 5 units.

Plotting the Graph of original function

Begin by graphing the original function \(f(x) = 3^x\). Some points to consider might be \((-1,1/3)\), \((0,1)\), and \((1,3)\). By plotting these points on a graph and connect them, you will have the graph of the function \(f(x)\).

Plotting the Graph of the transformed function

Now, graph the transformed function \(g(x) = 3^x + 5\). The '+5' means to take your original points from the function \(f(x)\) and add 5 to each of the y-values to get the points for \(g(x)\). i.e., the point \((-1,1/3)\) from \(f(x)\) will become \((-1, 5 + 1/3)\) for \(g(x)\). This process will be repeated for all points and the points are then plotted and joined to get the graph for \(g(x)\). You should see the graph of \(f(x)\), just shifted up by 5.

Key concepts

Exponential Functions

Exponential functions are a cornerstone in understanding a variety of scientific, mathematical, and financial models. An exponential function can be described by the general formula \( y = a^x \), where \( a \) is a positive constant base and \( x \) is the exponent. In the context of the given exercise, \( f(x) = 3^x \) is an exponential function with base 3, showing how the output value rapidly increases as \( x \) becomes larger.

Understanding the behavior of exponential functions is crucial as they exhibit unique characteristics: they pass through the point \( (0, 1) \) for any base \( a \) excluding 0, they grow without bound as \( x \) increases, and they approach zero as \( x \) becomes more negative. These functions are used to model phenomena like population growth, radioactive decay, and interest compounding where processes increase proportionally.

Graphing Transformations

Graphing transformations involve altering the appearance of a function's graph without changing its overall shape. These modifications can include shifts, stretches, compressions, and reflections. For our example with exponential functions, we focused on the 'vertical shift'.

A vertical shift occurs when we add or subtract a constant term to the function. In the exercise, adding 5 transforms \( f(x) \) into \( g(x) = f(x) + 5 \), effectively moving the entire graph of the original function up by 5 units on the y-axis. Graphing transformations is a handy tool in visualizing how equations change under certain modifications, and it also applies to other forms of functions beyond exponential.

Vertical Shifts

A vertical shift is one of the simplest forms of graph transformations. It translocates the entire graph of a function up or down the y-axis. In the problem provided, \( g(x) \) represents \( f(x) \) shifted vertically by 5 units upwards, as indicated by the \( +5 \) added to the original function's output. When we apply a vertical shift to a function, we do not alter the x-values of the points on the graph; only the y-values are increased or decreased by the same amount.

This concept is essential in understanding how to shift graphs without changing their shape, and it is especially helpful in adjusting models to fit new data or conditions.

Plotting Points

Plotting points is a fundamental skill in graphing and understanding functions. By calculating a set of ordered pairs \( (x, y) \) that satisfy a given function, we can plot these coordinates on a Cartesian plane and join them to visualize the function's graph. In our exercise with \( f(x) = 3^x \) and \( g(x) = 3^x + 5 \) as a transformed version, plotting points helps us see the impact of the transformation. To plot the transformed function \( g(x) \) accurately, you start with the points from \( f(x) \) and apply the vertical shift upwards by adding 5 to each y-coordinate, then plot these new points for \( g(x) \) on the graph.

Applying this process ensures the student not only grasps the visual difference between functions but also understands the underlying mathematical relationship guiding these changes.