Problem 103 - Elementary and Intermediate Algebra

Question

Graph each function by plotting points or by using a translation. The basic logarithmic functions graphed in Exercises \(99-102\) will be helpful. See Example 7 . $$ f(x)=3+\log _{3} x $$

Step-by-Step Solution

Answer

Graph \(f(x)=3+\log_{3}x\) by shifting \(\log_{3}x\) up by 3 units.

Step 1: Identify the Base Function

The core function here is the logarithmic function with base 3, specifically \(blog_3(x)\). This is our starting point for graphing.

Step 2: Translation of the Graph

The function \(f(x) = 3 + \log_{3}x\) represents a vertical translation of the basic function \(\log_{3}x\). The graph of \(\log_{3}x\) is shifted upwards by 3 units.

Step 3: Determine Key Points of \(\log_{3}x\)

For \(\log_{3}x\), characteristic points include \((1,0)\), \((3,1)\), and \((\frac{1}{3}, -1)\). These points come from evaluating the function where \(x = 3^n\) for integer \(n\).

Step 4: Apply the Translation

Shift each key point of \(\log_{3}x\) upwards by 3 units. This means the point \((1,0)\) becomes \((1,3)\), the point \((3,1)\) becomes \((3,4)\), and the point \((\frac{1}{3}, -1)\) becomes \((\frac{1}{3}, 2)\).

Step 5: Draw the Graph

Plot the shifted points \((1,3)\), \((3,4)\), and \((\frac{1}{3}, 2)\). Connect these points smoothly, ensuring the curve passes through them and reflects the logarithmic shape, which approaches negative infinity as \(x\) approaches 0 from the right.

Key Concepts

Translation of Graphs

When you're graphing a logarithmic function like \(f(x) = 3 + \log_{3}x\), it's essential to understand what the translation of the graph means. Translation refers to shifting the entire graph of a function in a particular direction without changing its shape or orientation. For the function given, the operation "\(+3\)" affects the entire graph. Here's how:

• The "3" adds a vertical shift. This moves the graph upwards by three units.
• The original graph of \(\log_{3}x\) has a distinct shape, approaching infinity to the right and negative infinity near 0. This shape remains unchanged except for the vertical shift.
• Key points on the original graph—like \((1,0), (3,1), (\frac{1}{3}, -1)\)—are shifted up. For example, \((1,0)\) becomes \((1,3)\), aligning the vertical increase.

Applying translation helps in analyzing how the original function adjusts in position while helping to maintain its key characteristics.

Logarithmic Functions

A logarithmic function, such as \(\log_{3}x\), plays a vital role in understanding the behavior of the given function \(f(x) = 3 + \log_{3}x\). Let's dive a bit deeper into what these functions are and how they work:

• A logarithmic function is the inverse of an exponential function. If \(3^y = x\), then \(y = \log_{3}x\).
• Graphically, it is depicted with a curve that rises rapidly, tapering off as \(x\) increases, remaining confined to positive values for \(x\).
• For \(\log_{3}x\), as \(x\) approaches zero from the right, the graph goes to negative infinity. In contrast, as \(x\) increases, the graph climbs upwards but slows down its ascent significantly.
• Logarithmic graphs are crucial in many applications, including solving equations related to compound interest, measuring sound levels, and determining growth time in biology.

Understanding logarithmic functions allows us to better navigate how alterations to their equations, like the vertical shift seen in \(f(x) = 3 + \log_{3}x\), unfold visually on their graphs.

Base of Logarithms

In logarithmic functions, the base essentially determines the rate at which the function increases. In this particular equation, \(f(x) = 3 + \log_{3}x\), the base is "3." Here's a closer look at how the base affects the function:

• The base "3" implies that the function \(\log_{3}x\) will intercept points where \(x = 3^n\) (where \(n\) is any integer) at integer values on the graph.
• For example, \(\log_{3}3 = 1\) because \(3^1 = 3\). Similarly, \(\log_{3}9 = 2\) because \(3^2 = 9\).
• Changing the base of a logarithm would affect how fast or slow the curves rise visually. A larger base results in a flatter graph, while a smaller base gives a steeper incline.
• Understanding the base is crucial for predicting behavior and applying logarithms in real-world scenarios. It directly impacts the scale and provides context for translating the graph effectively.

Hence, when graphing a translated logarithmic function, recognizing the base helps set expectations for how the function will behave across its domain.