Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graph the exponential function by hand. Identify any asymptotes and intercepts and determine whether the graph of the function is increasing or decreasing. $$f(x)=\left(\frac{3}{2}\right)^{-x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=(\frac{3}{2})^{-x}\) is decreasing, with a horizontal asymptote at \(y=0\) and a y-intercept at (0,1).

Step 1: Identify the Base and Exponent

In this function, \(f(x)=(\frac{3}{2})^{-x}\), the base is \(\frac{3}{2}\) and the exponent is \(-x\). This indicates that the function is decreasing, as the exponent is negative. When the base of the exponential function is greater than 1 and the exponent is negative, the function will decrease.

Step 2: Identify the Asymptotes

The horizontal asymptote for the exponential function is \(y=0\). This is because as \(x\) approaches infinity, the value of \(f(x)\) approaches 0 due to the negative exponent.

Step 3: Identify the Intercepts

The exponential function \(f(x)=(\frac{3}{2})^{-x}\) intersects the y-axis at the point (0,1), because any number raised to the power of zero yields 1. It does not, however, intersect the x-axis; there are no values for which \(f(x)=0\).

Step 4: Plotting the Function

Begin by plotting the y-intercept at (0,1). Because the function is decreasing, the curve will descend from the y-intercept towards the x-axis, getting closer but never touching it, due to the asymptote at \(y=0\). In the negative x direction, the curve will ascend, again stemming from the y-intercept at (0,1).

Key Concepts

Graphing Functions

Graphing an exponential function involves plotting its curve on a coordinate plane. Exponential functions can take various forms, but they usually include a base raised to a variable exponent. For the given function, \( f(x) = \left(\frac{3}{2}\right)^{-x} \), we begin by identifying the base \( \frac{3}{2} \) and the exponent \( -x \). These features dictate the shape and direction of the graph.

• The graph of an exponential function can rise or fall sharply based on its base and exponent.
• It's important to note that as \( x \) moves from negative to positive, or vice versa, the function's behavior is defined by whether the exponent is positive or negative.

With our function, the process begins by plotting key points, then sketching the smooth curve that connects them, following the trait of decrease as dictated by the negative exponent.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They are crucial in the study of exponential functions because they provide insights into the graph's behavior in extreme values of \( x \).
For the function \( f(x) = \left(\frac{3}{2}\right)^{-x} \), there's a horizontal asymptote at \( y = 0 \). This asymptote indicates that as \( x \) approaches infinity, the value of \( f(x) \) approaches but does not reach zero. This is typical for exponential functions with negative exponents, as they "flatten out" near zero when moving positively along the x-axis. Understanding asymptotes helps us in correctly drawing and interpreting the overall shape of the graph.

Intercepts

Intercepts are points where the graph crosses the axes. They provide a snapshot of initial values for a function when the variables are zero, which can be particularly useful for graphing.
For the function \( f(x) = \left(\frac{3}{2}\right)^{-x} \):

• **Y-Intercept:** This function has a y-intercept at \( (0, 1) \). Any base raised to the power of zero equals one, making \( f(0) = 1 \).
• **X-Intercept:** There is no x-intercept for this function. Exponential functions of the form \( b^{-x} \), where \( b \) is positive, never actually cross the x-axis, meaning \( f(x) eq 0 \) for any real number \( x \).

These intercepts are integral when sketching the function, as they offer the initial anchor points from which a graph can be drawn.

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing is critical when analyzing its graph. For exponential functions, the base and sign of the exponent are key determinants.
With \( f(x) = \left(\frac{3}{2}\right)^{-x} \), the function is decreasing. Here's why:

• **Base Greater Than 1 and Negative Exponent:** When an exponential function has a base greater than 1 but a negative exponent, the function decreases as \( x \) increases.
• This exponential decay results in a curve that moves downward from left to right on a graph, starting from the y-axis (which acts as an anchor point here).

In this context, it's easy to visualize how the curve descends towards the horizontal asymptote at \( y = 0 \), making for a clear representation of a decreasing trend as \( x \) progresses.