Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=\left(\frac{2}{3}\right)^{-x}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = \left(\frac{3}{2}\right)^x\) is an increasing exponential function with a horizontal asymptote at \(y=0\).

Step 1: Identify the Base Function

The given function is \(f(x) = \left(\frac{2}{3}\right)^{-x}\). Notice that the base function before transformation is \(g(x) = \left(\frac{2}{3}\right)^{x}\).

Step 2: Understand the Negative Exponent Effect

The function \(f(x) = \left(\frac{2}{3}\right)^{-x}\) can be rewritten using the property of exponents: \(\left(a^{-b}\right) = \left(\frac{1}{a}\right)^{b}\). Therefore, \(f(x) = \left(\frac{3}{2}\right)^{x}\). This implies that the graph will be an exponential function with a base greater than 1, which results in a growth trend.

Step 3: Determine Key Points

To graph the function accurately, calculate a few key points. For easy computations, choose values like: - \(x=0\): \(f(0) = \left(\frac{3}{2}\right)^0 = 1\)- \(x=1\): \(f(1) = \left(\frac{3}{2}\right)^1 = \frac{3}{2}\)- \(x=-1\): \(f(-1) = \left(\frac{3}{2}\right)^{-1} = \frac{2}{3}\)These points help in drawing the curve.

Step 4: Determine the Asymptote

As with all exponential functions, \(f(x) = \left(\frac{3}{2}\right)^x\) has a horizontal asymptote. As \(x\rightarrow -\infty\), \(f(x)\) approaches 0, which is the horizontal asymptote on the x-axis.

Step 5: Sketch the Graph

Using the points \((0, 1)\), \((1, \frac{3}{2})\), and \((-1, \frac{2}{3})\), and understanding that the function is exponentially increasing (due to the base \(\frac{3}{2}\) being greater than 1), sketch the curve. Start from the asymptote at \(y = 0\) on the left and draw a curve up through the plotted points, increasing to the right.

Key Concepts

Graphing Exponential Functions

Exponential functions are an essential part of mathematics, and graphing them helps visualize their behavior. An exponential function generally takes the form \(y = a^x\), where \(a\) is the base. For base values greater than 1, such functions exhibit exponential growth, while bases between 0 and 1 lead to decay.
Understanding how to sketch these functions involves recognizing their general shape. Exponential growth functions rapidly increase as \(x\) increases, moving upwards sharply in one direction. To begin graphing, it's advisable to identify a few key points.

• Start at \(x=0\), where the function usually equals 1 if no other transformations are applied
• Choose additional points, like \(x = 1\) and \(x = -1\), to understand the curvature.

These points can then be plotted, using the typical J-shaped curve for a base greater than 1. Remember, the graph will always pass on one side of the vertical line through \(x=0\), determined by the base value.

Negative Exponents

Negative exponents in mathematical expressions often confuse students, yet they follow simple rules. A crucial property of negative exponents is that they represent reciprocal values. If you ever see an expression like \(a^{-b}\), interpret it as \(\left(\frac{1}{a}\right)^b\).
This transformation is evident in the function \(f(x) = \left(\frac{2}{3}\right)^{-x}\). By rewriting it, we transform the base to a positive exponent, yielding \(f(x) = \left(\frac{3}{2}\right)^x\). This inversion not only simplifies calculation but alters the function's behavior:

• Wherever you have decay (base less than 1), a negative exponent flips it to growth.
• Understanding this switch is crucial for correct function interpretation.

Always remember to rewrite functions with negative exponents into their equivalent form to easily predict their graphs.

Horizontal Asymptotes

Horizontal asymptotes are lines that an exponential function approaches but never actually reaches. They are crucial for understanding the limits of a function's behavior without having to compute numerous points.
In the context of exponential functions, horizontal asymptotes usually appear on the x-axis (\(y=0\)). Knowing where this line is allows students to foresee the function's end behavior as \(x\) approaches negative or positive infinity.

• For exponential growth (base greater than 1), the graph will move away from the horizontal asymptote as \(x\) increases.
• During decay, the function moves closer to the asymptote as \(x\) increases.

For \(f(x) = \left(\frac{3}{2}\right)^x\), note how the curve drops towards the x-axis as \(x\rightarrow -\infty\), highlighting the horizontal asymptote at \(y=0\). Understanding this behavior allows the graph to be more accurately depicted.

Transformation of Functions

Transformation of functions involves altering the original function into a new one either through shifts, reflections, or stretches/compressions. For exponential functions, such transformations can drastically change their appearance without altering their general form.
Let's consider the function \(f(x) = \left(\frac{2}{3}\right)^{-x}\). We see a transformation happening with the negative sign in the exponent: this reflects the graph over the y-axis. Transformations can include:

• Vertical or horizontal shifts, moving the graph up, down, or sideways.
• Vertical reflections, as seen with negative exponents.
• Vertical stretches or compressions, affecting the steepness of the graph.

By recognizing and applying these transformations, you can predict how a function will behave under various changes, enhancing your graphing skills and understanding of exponential functions.