Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$f(x)=3^{x}-3^{-x}$$

Step-by-Step Solution

Answer

Sketch involves symmetry around origin with the curve passing through calculated points.

Step 1: Identify the Base Function

The function given is \(f(x) = 3^x - 3^{-x}\). Here, \(3^x\) is an exponential growth function and \(3^{-x}\) is an exponential decay function. The graph will involve combining these behaviors.

Step 2: Compute Key Points

Calculate \(f(x)\) for key values: 1. \(f(0) = 3^0 - 3^{-0} = 1 - 1 = 0\)2. \(f(1) = 3^1 - 3^{-1} = 3 - \frac{1}{3} = \frac{8}{3}\)3. \(f(-1) = 3^{-1} - 3^1 = \frac{1}{3} - 3 = -\frac{8}{3}\)4. \(f(2) = 3^2 - 3^{-2} = 9 - \frac{1}{9} = \frac{80}{9}\)5. \(f(-2) = 3^{-2} - 3^2 = \frac{1}{9} - 9 = -\frac{80}{9}\)These points will help shape the graph.

Step 3: Analyze Symmetry and Asymptotes

Notice the function is odd: \(f(-x) = -f(x)\), indicating symmetry about the origin. Since there are no terms causing horizontal asymptotes, and as \(x\) approaches infinity, \(3^x\) dominates, making the graph rise steeply in quadrant I.

Step 4: Sketch the Graph

Using the key points and symmetry, plot \((0, 0)\), \((1, \frac{8}{3})\), \((-1, -\frac{8}{3})\), \((2, \frac{80}{9})\), and \((-2, -\frac{80}{9})\). Draw a curve through these points considering that the graph is steep due to the exponential terms. The graph should be in the first and third quadrants, rising in the first quadrant and falling in the third quadrant symmetrically.

Key Concepts

Exponential Growth and Decay

In the given function, we have two main components: exponential growth and exponential decay. Understanding these concepts is crucial as they form the foundation of the function's behavior.

**Exponential Growth**: This occurs when a quantity increases at a consistent rate over time. In our function, the term \(3^x\) is responsible for this behavior. As \(x\) becomes larger, \(3^x\) increases very rapidly, leading the graph to rise sharply.

**Exponential Decay**: This refers to a quantity decreasing at a consistent rate over time. The term \(3^{-x}\) exhibits exponential decay. As \(x\) grows, \(3^{-x}\) shrinks towards zero.

Together, these components interact within the function \(f(x) = 3^x - 3^{-x}\). When \(f(x)\) is considered, both growth and decay factors influence the outcome, creating a distinctive curve as we sketch the graph.

Symmetry in Functions

Symmetry is a helpful property that makes graphing functions easier.

For many functions, symmetry provides clues about their geometric properties. In our function, \(f(x) = 3^x - 3^{-x}\), there is symmetry about the origin.

**Origin Symmetry**: A function is origin symmetric if \(f(-x) = -f(x)\). This implies that the graph will mirror itself in the opposite quadrant, making our graph have a reflection across the term \(x = 0\).

In this particular function, when we substitute \(-x\) into the function, we find \(f(-x) = -f(x)\), confirming origin symmetry.

Understanding this property allows us to know that if the graph passes through a point in one quadrant, its corresponding point will appear in the opposite quadrant. This symmetry helps simplify graph sketching.

Key Points in Graph Sketching

Identifying key points in a function plays an essential role in accurately sketching its graph. These points serve as a roadmap, indicating how the graph behaves over the range of \(x\) values.

For our function, computing \(f(x)\) at strategic positions provides these critical points:

• At \(x=0\), \(f(0) = 0\) establishes the intersection of the graph with the origin.


• At \(x=1\), \(f(1) = \frac{8}{3}\) indicates the function is already climbing, showcasing exponential growth.


• At \(x=-1\), \(f(-1) = -\frac{8}{3}\) reflects the point because of the symmetry, descending in the third quadrant.


• At \(x=2\), \(f(2) = \frac{80}{9}\) further showcases how the function surges upward steeply.


• At \(x=-2\), \(f(-2) = -\frac{80}{9}\) again reflects due to symmetry.

These points assist us in plotting the graph correctly and understanding the exponential tendencies. By connecting these dots with a smooth curve, you define the trajectory and nature of the function's graph.