Problem 48 - Algebra and Trigonometry

Question

Draw the graph of \(y=3^{x},\) then use it to draw the graph of \(y=\log _{3} x .\)

Step-by-Step Solution

Answer

Graph \(y=3^x\) first, then reflect it over \(y=x\) to get \(y=\log_{3} x\).

Step 1: Draw the Graph of y=3^x

First, consider the function \(y = 3^x\). This is an exponential function with base 3. To sketch the graph, note the following key points: when \(x = 0\), \(y = 3^0 = 1\); when \(x = 1\), \(y = 3^1 = 3\); and when \(x = -1\), \(y = 3^{-1} = \frac{1}{3}\). Draw the curve passing through these points, rising from left to right as \(x\) increases, approaching zero as \(x\) decreases.

Step 2: Understand the Relationship y=log_3 x

The graph of \(y = \log_{3} x\) is the inverse of \(y = 3^x\). For inverse functions, the graph is a reflection of the original function across the line \(y = x\). This means any point \((a, b)\) on \(y = 3^x\) corresponds to a point \((b, a)\) on \(y = \log_{3} x\). For instance, \((0, 1)\) on \(y=3^x\) corresponds to \((1, 0)\) on the logarithmic graph.

Step 3: Draw the Graph of y=log_3 x

Reflect the points from the graph of \(y=3^x\) to plot those on \(y = \log_{3} x\): \((1, 0)\), \((3, 1)\), \(\left(\frac{1}{3}, -1\right)\). Connect these dots smoothly, creating a curve that continues to rise slowly as \(x\) increases and approaches negative infinity as \(x\) approaches zero from the right. Complete the reflection to create the full \(y=\log_{3} x\) graph, which is now sketched.

Key Concepts

Exponential Functions

Exponential functions are a fundamental concept in algebra and calculus. They are defined as functions where a constant base is raised to a variable exponent. For example, in the function \(y = 3^x\), 3 is the base, and \(x\) is the exponent. This type of function is known for its growth properties.

• When the base is greater than one, the function grows rapidly as the exponent increases.
• If the base is between zero and one, the function decreases as the exponent increases.

To grasp an exponential function fully, it's important to observe its behavior through key points. Like in our example, for \(x = 0\), \(y\) equals 1; for \(x = 1\), \(y\) equals the base; and for \(x = -1\), \(y\) diminishes, resulting in a fraction. Viewing these points help understand how the curve behaves, starting with a flat approach near zero and quickly rising as \(x\) increases.

Inverse Functions

Inverse functions provide a unique perspective in mathematics, essentially flipping the coordinates of the points on a graph. If you have a function \(f(x)\), its inverse, denoted \(f^{-1}(x)\), will give you \(x\) when \(f(x) = y\). The concept is like solving for the original input given the output.

• One key aspect is that the graph of an inverse function is the reflection of the original function over the line \(y=x\).
• This means a point \((a, b)\) on the function becomes \((b, a)\) on its inverse.

In our specific case, the exponential function \(y = 3^x\) and its inverse logarithmic function \(y = \log_{3} x\) demonstrate this beautifully. The inverse is visually represented by swapping the coordinates of its exponential counterpart. This reflection reveals how inputs in one function correspond to outputs in the other, providing a powerful tool for understanding relationships between quantities.

Graphing Techniques

When it comes to graphing functions, there are useful techniques you can employ to get an accurate visual representation. It starts with plotting key points and understanding how they connect to form the curve.For the exponential function \(y = 3^x\), select points like \((0, 1)\), \((1, 3)\), and \((-1, \frac{1}{3})\). Draw a smooth curve passing through these, demonstrating exponential growth. This curve rises sharply from left to right.To graph its inverse, the logarithmic function \(y = \log_{3} x\), reflect these points over the line \(y = x\). This involves swapping the coordinates: \((1, 0)\), \((3, 1)\), and \((\frac{1}{3}, -1)\). Connect these points to form a slowly rising curve, illustrating how it approaches infinity as \(x\) increases and negative infinity as \(x\) nears zero from the right.These methods ensure clarity, making graphing both informative and understandable. Always ensure to track the shifts and reflections accurately when transitioning from one graph to its inverse, providing a seamless lesson in visualization.