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Magazine Chapter 5: Problem 52
Draw the graph of \(y=3^{x}\) then use it to draw the graph of \(y=\log _{3} x\)
Short Answer
Expert verified
Reflect the exponential graph over y=x to get the logarithmic graph.
Step by step solution
01
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 \(x = -2, -1, 0, 1, ext{ and } 2\).
02
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.
03
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.
04
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.
05
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.
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
