Problem 35 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=2^{-x}-1$$

Step-by-Step Solution

Answer

The graph begins at the point (0, 0), falls gradually towards \( y=-1 \) as \( x \) becomes larger, and rises sharply as \( x \) becomes negative. The asymptote of this graph is \( y=-1 \).

Step 1: Understanding the given function

The given function \(f(x)=2^{-x}-1\) is an exponential function because it has a base of 2 raised to the power of \( -x \). The '-1' is a shift transformation. It shifts the exponential function downward by one unit.

Step 2: Determine the graph behavior

In general, the function \(2^{-x}\) shows exponential decay as the exponent becomes more negative. However, the function decreases as \( x \) increases. For \( x=0 \), the graph passes through the point (0,1), but with the '-1' shift, this point will be (0,0). The function approaches \( y=-1 \) as \( x \) goes to infinity.

Step 3: Sketch the graph

Begin by plotting the y-intercept at (0, 0). As \( x \) increases, the graph approaches the horizontal asymptote at \( y=-1 \). As \( x \) decreases, \( y \) should rise sharply. We can also choose some values for \( x \) to put into the function to get some corresponding values for \( y \), and then we can plot these points, following the general behavior and the information in previous steps.

Key Concepts

Graph Sketching

Graph sketching is an essential skill that helps visualize mathematical functions. When sketching the graph of an exponential function like \( f(x) = 2^{-x} - 1 \), it's important to note key points and behaviors.

• Identify intercepts: For this function, when \(x = 0\), \( f(0) = 2^{0} - 1 = 0\). So, the graph passes through the point (0,0).
• Understand asymptotic behavior: As \( x \) approaches infinity, \( f(x) \) gets closer to its horizontal asymptote at \( y = -1 \).
• Choose additional points: Substitute values for \( x \) to estimate more points, like \( x = 1 \) giving \( f(1) = \frac{1}{2} - 1 = -\frac{1}{2} \), and plot these on the graph.
• Consider symmetry and trends: The graph of \( f(x) = 2^{-x} - 1 \) will decay since the base of the exponent is less than 1 due to the negative power.

Plot these points and smoothly join them to form the curve. Respect the asymptotes as limits the graph never crosses.

Exponential Decay

Exponential decay occurs when a function reduces towards a baseline as the input grows. For the function \( f(x) = 2^{-x} - 1 \), the base is part of what causes this decay.

• Exponential nature: Here, the function \( 2^{-x} \) decays because the exponent \(-x\) inversely affects the growth. As x becomes positive, \( 2^{-x} \) becomes a fraction, decreasing overall value.
• Decay behavior: The graph tends toward its asymptote \( y = -1 \) since \( 2^{-x} + c \) approaches \( 0 + c \) as \( x \to \infty \).
• Real-world context: This kind of decay often models things like radioactive decay or cooling temperatures, where quantities decrease rapidly at first then stabilize.

Understanding decay helps in visualizing the graph correctly to see how the function ebbs over increasing \( x \).

Transformations

Transformations modify functions to shift, stretch, or compress their graphs. For \( f(x) = 2^{-x} - 1 \), specific transformations make the graph distinctive.

• Vertical shifts: The \(-1\) in \( f(x) = 2^{-x} - 1 \) shifts the whole graph downwards by 1 unit, altering the baseline position.
• Horizontal transformations: The negative sign in the exponent \(-x\) flips the graph left to right compared to a typical exponential growth function.
• Transformation impact: The combined result is a graph that decays (goes down) starting from (0,0), higher up then peels away toward \( y = -1 \) moving right.

Applying these transformations influences the graph's behavior and starting points, exemplifying how mathematical changes alter visual representations.