Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=2^{-|x|}$$

Step-by-Step Solution

Answer

The graph of \(f(x) = 2^{-|x|}\) is even, symmetrical about the y-axis, and has a horizontal asymptote at \(y=0\).

Step 1: Identify the function's form

The function given is \( f(x) = 2^{-|x|} \). Here, \(|x|\) denotes the absolute value of \(x\), so the expression \(-|x|\) ensures the exponent is non-positive.

Step 2: Analyze symmetry

Since the function involves \(|x|\), it is even, meaning \(f(x) = f(-x)\). This implies the graph will be symmetric with respect to the y-axis.

Step 3: Determine key points

Calculate values for key points:- \(f(0) = 2^{0} = 1\)- \(f(1) = 2^{-1} = \frac{1}{2}\)- \(f(-1) = 2^{-1} = \frac{1}{2}\)Continue this pattern for other integer values to sketch more points accurately.

Step 4: Behavior as \(x\) approaches extremes

As \(x\) approaches positive or negative infinity, \(|x|\) increases, making \(-|x|\) more negative, and \(2^{-|x|}\) approaches 0. Therefore, the graph approaches the x-axis but never touches it, having a horizontal asymptote at \(y=0\).

Step 5: Sketch the graph

Plot the points calculated in Step 3 on the coordinate axes. Ensure symmetry with respect to the y-axis, and draw a curve that starts at (0,1), passing through calculated points, and approaches the x-axis as \(x\) moves away from zero.

Key Concepts

Graph Symmetry

Understanding graph symmetry is crucial when sketching functions like \(f(x) = 2^{-|x|}\). A graph is symmetric with respect to the y-axis if for every point \((x, y)\) on the graph, the point \((-x, y)\) is also on the graph. This is typical for even functions.

Since \(f(x) = 2^{-|x|}\) uses an absolute value in its exponent, the function mirrors itself over the y-axis. Whether \(x\) is positive or negative, the absolute value makes no difference to the outcome as it's always positive, hence \(f(x) = f(-x)\).

This y-axis symmetry significantly simplifies graphing because you need to calculate values for only half of the \(x\)-axis and can reflect them on the other side. It's a helpful property when sketching or analyzing graphs.

Absolute Value Functions

The function \(f(x) = 2^{-|x|}\) features the absolute value of \(x\), which is a crucial component in mathematics. An absolute value function transforms any input into its non-negative equivalent.

In the given function, the absolute value is used in the exponent \(|x|\). The absolute value has a fascinating property: it makes the function behave identically for positive and negative inputs. This results in the even symmetry we discussed previously.

Absolute value functions often create V-shaped graphs when they stand alone (e.g., \(f(x) = |x|\)). However, in our context with \(2^{-|x|}\), the absolute value leads to a unique exponential decay, offering a distinct visual pattern as it causes the curve to decrease symmetrically.

Horizontal Asymptotes

The concept of horizontal asymptotes is essential in understanding the long-term behavior of functions. An asymptote is a line that a graph approaches but never actually touches. In \(f(x) = 2^{-|x|}\), there's a horizontal asymptote at \(y=0\).

As \(x\) becomes very large or very small, the absolute value \(|x|\) increases, making the exponent \(-|x|\) decrease. Consequently, \(2^{-|x|}\) becomes a fraction that shrinks to zero. This behavior is why the graph gets closer and closer to the x-axis as \(x\) moves towards infinity in both directions.

Horizontal asymptotes are crucial for predicting the end-behavior of a graph. They help us understand how the graph behaves as \(x\) approaches positive or negative infinity, making them invaluable for analyzing and sketching exponential functions.