Problem 22 - Precalculus : Building Concepts and Connections

Question

Sketch the graph of each function. $$f(x)=4(2)^{-x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = 4(2)^{-x}\) starts at the y-intercept (0,4). As \(x\) gets larger (moving to the right), the graph gets closer to the x-axis but never touches it, indicating \(f(x)\rightarrow 0\) as \(x\rightarrow \infty\). And as \(x\) gets smaller (moving to the left), the graph increases without bound, indicating \(f(x)\rightarrow \infty\) as \(x\rightarrow -\infty\).

Step 1: Identify the y-intercept

The y-intercept is the point where the function crosses the y-axis. This occurs when \(x = 0\). Substituting \(x = 0\) in the function, \(f(x) = 4(2)^{-x}\), we get \(f(0) = 4(2)^0 = 4\). Thus, the y-intercept is at point (0,4).

Step 2: Determine the behavior as \(x\rightarrow \infty\)

As \(x\) approaches infinity (\(x\rightarrow \infty\)), \(2^{-x}\) becomes very small (approaches 0) because any number (except 0) raised to the power of negative infinity is 0. Therefore, \(f(x)\rightarrow 0\) as \(x\rightarrow \infty\). The graph will get asymptotically close to the x-axis but never touch or cross it.

Step 3: Determine the behavior as \(x\rightarrow -\infty\)

As \(x\) approaches negative infinity (\(x\rightarrow -\infty\)), \(2^{-x}\) becomes very large (approaches infinity) because any number (except 0) raised to the power of infinity is infinity. Therefore, \(f(x)\rightarrow\infty\) as \(x\rightarrow -\infty\). The graph will rapidly increase without bound as we move to the left of the y-axis.

Step 4: Plot the function

Plot the y-intercept from Step 1 and use the information from Step 2 and Step 3 to sketch the graph of \(f(x) = 4(2)^{-x}\). The graph starts at the y-intercept (0,4) and declines towards the x-axis as \(x\) gets larger (moving to the right), but gets closer to positive infinity as \(x\) gets smaller (moving to the left).

Key Concepts

Understanding the Y-intercept

When working with exponential functions, the y-intercept is a crucial point to identify. This is where the graph of the function crosses the y-axis. It represents the output value of the function when the input variable, usually denoted by \(x\), is equal to zero. In the exercise, the function given is \(f(x) = 4(2)^{-x}\). To find the y-intercept, substitute \(x = 0\) into the equation. This gives:

• \(f(0) = 4(2)^0\)
• \(2^0\) equals 1, so \(f(0) = 4 \times 1 = 4\).
• Therefore, the y-intercept is (0, 4).

The y-intercept can serve as an anchoring point when sketching the graph, giving a starting point to visualize how the curve behaves along the y-axis. Because this function is a decreasing exponential function due to the negative exponent, the graph will curve downwards from this point as \(x\) increases.

Exploring Asymptotic Behavior

Asymptotic behavior in the context of exponential functions refers to how the function approaches a line or a value without actually reaching it. For the function \(f(x) = 4(2)^{-x}\), the graph has a horizontal asymptote along the x-axis. This means that as \(x\) increases, or moves towards positive infinity, the value of the function approaches zero. However, it will never actually reach zero. Let's delve into why:

• The term \(2^{-x}\) results in very small numbers as \(x\) becomes large.
• This is because raising a number greater than one to a negative exponent makes the result a fraction.
• As \(x\) approaches infinity, the value \(2^{-x}\) approaches zero.
• Thus, \(f(x) = 4 \times 2^{-x}\) also approaches zero.

This behavior is called asymptotic because the graph gets infinitely close to the x-axis without ever crossing it. Recognizing this behavior helps to accurately sketch and understand how the graph "flattens out" towards an imaginary line.

Infinity Behavior of the Graph

The behavior of a graph as \(x\) approaches positive or negative infinity can provide significant insights into its nature. With exponential functions like \(f(x) = 4(2)^{-x}\), infinity behavior explains how the graph behaves at the extremes. Looking at both sides:

• As \(x\rightarrow \infty\), the function approaches zero, which we've discussed under asymptotic behavior.
• As \(x\rightarrow -\infty\), however, the behavior changes substantially.

On this side, the negative exponent \(-x\) inverts the base, leading to a large number. Essentially:

• \(2^{-x}\) becomes very large as \(x\) approaches negative infinity.
• This is because \(-x\) with a negative value means the exponent will be positive.
• Thus \(f(x) = 4(2)^{-x}\) becomes very large, tending towards infinity.

This results in the graph rising steeply as \(x\) decreases, highlighting how exponential growth or decay impacts graphical interpretation. Understanding this dual behavior of the function at extreme values of \(x\) is vital for accurate graphing and analysis.