Problem 52 - 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

Reflect the exponential graph over y=x to get the logarithmic graph.

Step 1: Understand the function and choose points

The function we need to graph first is an exponential function, \(y = 3^x\). An exponential function has the general form \(y = a^x\), where the base, in this case, is \(a = 3\). To draw its graph, we choose a few values of \(x\) for which we can calculate \(y\) easily, such as \(x = -2, -1, 0, 1, ext{ and } 2\).

Step 2: Plot the points for the graph of y = 3^x

Calculate the values of \(y = 3^x\) for the chosen points: - \(x = -2: y = 3^{-2} = \frac{1}{9}\)- \(x = -1: y = 3^{-1} = \frac{1}{3}\)- \(x = 0: y = 3^0 = 1\)- \(x = 1: y = 3^1 = 3\)- \(x = 2: y = 3^2 = 9\)After calculating these, you can plot the points \((-2, \frac{1}{9}), (-1, \frac{1}{3}), (0, 1), (1, 3), (2, 9)\) on a coordinate plane.

Step 3: Draw the graph of y = 3^x

Using the points plotted in the previous step, draw a smooth curve through these points to represent the exponential function \(y = 3^x\). This graph will increase rapidly as \(x\) increases and approach zero as \(x\) becomes more negative, but will never touch the x-axis.

Step 4: Understand the properties of the inverse function

The function \(y = \log_{3}x\) is the inverse of \(y = 3^x\), meaning the graphs of these functions are reflections of each other over the line \(y = x\). This line acts as a mirror for the functions to reflect upon.

Step 5: Reflect the points over the line y = x

To draw the graph of \(y = \log_{3}x\), we take each point from the graph \(y = 3^x\) and reflect it over the line \(y = x\). The original points \((-2, \frac{1}{9}), (-1, \frac{1}{3}), (0, 1), (1, 3), (2, 9)\) are transformed into \((\frac{1}{9}, -2), (\frac{1}{3}, -1), (1, 0), (3, 1), (9, 2)\) respectively.

Step 6: Draw the graph of y = log_{3}x

Using the reflected points: \((\frac{1}{9}, -2), (\frac{1}{3}, -1), (1, 0), (3, 1), (9, 2)\), plot these on the coordinate plane. Connect the points with a smooth curve. The logarithmic graph will increase slowly as \(x\) increases, never becoming horizontal, and is vertical as \(x\) approaches zero from the positive side but never crossing the y-axis.

Key Concepts

Exponential functions

Exponential functions are a fundamental concept in mathematics, commonly taking the form \(y = a^x\). These functions are characterized by a constant base \(a\) and a variable exponent \(x\). For instance, when graphing the exponential function \(y = 3^x\), we see how the function behaves:

• As \(x\) increases, \(y\) increases rapidly, demonstrating exponential growth.
• As \(x\) becomes more negative, \(y\) approaches zero, but never quite reaches it, indicating there is a horizontal asymptote at \(y = 0\).
• At \(x = 0\), the value of \(y\) is always 1, regardless of the base \(a\), because any number raised to the power of 0 is 1.

You can sketch an exponential graph by choosing key points that make calculations simple. For example, by selecting points like \(-2, -1, 0, 1,\) and \(2\) for \(x\), you can easily compute values for \(y\). This helps in visualizing the steep growth and the approach towards the horizontal asymptote as \(x\) decreases.

Logarithmic functions

Logarithmic functions are the inverse operations of exponential functions, expressed generally as \(y = \log_{a}x\), where \(a\) is the base. The logarithmic function \(y = \log_3 x\), explored in this context, has several interesting properties:

• The function is undefined for \(x \leq 0\). This means the graph exists only in the first quadrant.
• The graph passes through the point \((1, 0)\), because \(\log_a 1 = 0\) for any base \(a\).
• As \(x\) increases, the function grows slowly compared to exponential functions. This is because the growth is logarithmic.
• It approaches, but never crosses, the y-axis, which acts as a vertical asymptote.

Understanding the relationship between these two types of functions and their graphs is crucial for interpreting mathematical data involving exponential growth and logarithmic scales.

Inverse functions

Inverse functions are pairs of functions that essentially "undo" each other. In our case, the exponential function \(y = 3^x\) and its inverse, the logarithmic function \(y = \log_3 x\), are perfect examples. For these types of functions:

• Reflecting the graph of one function about the line \(y = x\) gives the graph of its inverse. Essentially, if a point \((a, b)\) exists on the graph of the original function, then \((b, a)\) will be a point on the graph of the inverse function.
• They both share some symmetry properties related to this line of reflection, which helps when graphing or analyzing mathematical relationships.

This reflective property between exponential and logarithmic functions highlights the interconnectedness in their behavior and graphical representation.

Graphing techniques

Graphing techniques play a vital role in understanding the behavior of functions visually. For graphing both exponential and logarithmic functions, some effective strategies include:

• Selecting strategic points: Choose values for \(x\) that are easy to compute to plot points without difficulty. This step is crucial for drawing an accurate graph.
• Reflecting points: When graphing the inverse, use reflection over the line \(y = x\) to quickly obtain points for the inverse function's graph.
• Asymptotes and intercepts: Identify where the function intercepts the axes and note any asymptotic behavior, which guides the curve's path as \(x\) or \(y\) increases or decreases dramatically.

Armed with these techniques, students can confidently sketch and interpret complex graphs, enhancing their understanding and analysis skills in mathematics.