Problem 48 - College Algebra with Modeling and Visualization

Question

Sketch a graph of \(y=f(x)\) $$ f(x)=3\left(2^{-x}\right) $$

Step-by-Step Solution

Answer

The graph is a decreasing exponential curve approaching the \( x \)-axis as \( x \) increases.

Step 1: Identify the Function Type

The given function is an exponential function: \( f(x) = 3(2^{-x}) \). This type of function decreases as \( x \) increases because it has a negative exponent.

Step 2: Determine Key Points

Choose a few values of \( x \) to determine the corresponding \( y \) values: \( f(0) = 3(2^0) = 3 \), \( f(1) = 3(2^{-1}) = \frac{3}{2} \), \( f(-1) = 3(2^{1}) = 6 \), and \( f(2) = 3(2^{-2}) = \frac{3}{4} \).

Step 3: Analyze Asymptotic Behavior

As \( x \) approaches infinity, \( 2^{-x} \) approaches 0, so \( f(x) \) approaches the horizontal asymptote \( y = 0 \).

Step 4: Plot the Points

Plot the points \((0,3)\), \((1,1.5)\), \((2,0.75)\), and \((-1,6)\) on a Cartesian plane, showing the decreasing nature of the function.

Step 5: Draw the Sketch

Connect the points with a smooth curve, starting high on the left, decreasing while moving to the right, getting closer to \( y=0 \), but never touching it.

Key Concepts

Graph Sketching

When sketching the graph of an exponential function like \( f(x) = 3(2^{-x}) \), it's important to remember that the base and the exponent play key roles in determining the graph's features. The general form of an exponential function is \( a \cdot b^x \), where \( a \) is the initial value or the vertical stretch factor, and \( b \) is the base that dictates growth or decay. In our function, \( a = 3 \) and \( b = 2^{-1} \).

Steps to sketch the graph include:

• Select several values of \( x \) and calculate the corresponding \( y \) values for those points. For example, when \( x = 0 \), \( y = 3 \); when \( x = 1 \), \( y = 1.5 \); and so on.
• Plot these points on a Cartesian coordinate system to help visualize the graph.
• Connect these points smoothly to show the behavior of the function, noting how it decreases.

The graph should start high from the left and decrease quickly, leveling off towards the right as it approaches the x-axis without ever crossing it.

Asymptotic Behavior

The term "asymptotic behavior" refers to how a function behaves as it moves towards infinity or negative infinity. For an exponential decay function like \( f(x) = 3(2^{-x}) \), the significant asymptote is the horizontal line \( y = 0 \). This line represents a value that the function approaches but never actually reaches.

Understanding asymptotes is crucial because:

• They help determine the end behavior of the graph.
• They indicate leveling patterns as the function values get smaller with increasing or decreasing \( x \).

For our function, as \( x \to \infty \), \( 2^{-x} \to 0 \), causing \( f(x) \to 0 \). This means the graph gets closer and closer to the x-axis, illustrating the function’s asymptotic nature without crossing the line itself.

Function Transformation

Function transformation allows us to modify the basic form of functions and explore how these changes affect their graphs. For \( f(x) = 3(2^{-x}) \), this involves shifts, stretches, and reflections derived from changing parameters and exponents.

Key transformations are:

• Vertical Stretch: The factor \( 3 \) stretches the basic graph vertically, influencing the amplitude of the function.
• Reflection: The negative exponent \(-x\) reflects the base function \(2^x\) across the y-axis, turning it into a decay function.

By understanding these transformations, you can predict how the graph will change: it will decrease more sharply thanks to the reflection and vertical stretch, starting from a higher point due to the vertical scaling.