Problem 103 - Introductory and Intermediate Algebra for College Students 4th

Question

a. Use a graphing utility (and the change-of-base property) to graph \(y=\log _{3} x\) b. Graph \(\quad y=2+\log _{3} x, \quad y=\log _{3}(x+2), \quad\) and \(y=-\log _{3} x \quad\) in the same viewing rectangle as \(y=\log _{3} x .\) Then describe the change or changes that need to be made to the graph of \(y=\log _{3} x\) to obtain each of these three graphs.

Step-by-Step Solution

Answer

To obtain \(y=2+\log_{3}x\), the base graph needs to be shifted 2 units upward; to occur \(y=\log_{3}(x+2)\), the base graph needs to be shifted 2 units to the left; to plot \(y=-\log_{3}x\), the base graph needs to be flipped over the x-axis.

Step 1: Graphing the Base Function

Using a graphing utility and using the change-of-base property, the graph of \(y=log_{3}x\) is plotted. The graph would rise slowly and approach the x-axis without touching or crossing it.

Step 2: Addition to the Function

Next, we plot \(y=2+log_{3}x\). Addition of 2 to the function indicates a vertical shift. It means that this graph is simply the graph of \(y=log_{3}x\) shifted 2 units upward.

Step 3: Addition Inside the Logarithm Function

Now, the graph for \(y=\log_{3}(x+2)\) is plotted. The equation indicates that there is a horizontal shift of the base graph. This graph is the graph of \(y=log_{3}x\) shifted 2 units to the left.

Step 4: Negation of the Base Function

Finally, the graph of \(y=-\log_{3}x\) is plotted. The negative sign before the logarithm function flips the base graph over the x-axis.

Step 5: Summarizing the Transformation

The following changes are made to the base graph: a. \(y=2+\log_{3}x\) - Base graph shifted 2 units up; b. \(y=\log_{3}(x+2)\) - Base graph moved 2 units to the left; c. \(y=-\log_{3}x\) - Base graph flipped across the x-axis.

Key Concepts

Graphing Transformations

Graphing transformations are techniques used to visually represent changes to a base function. A base function like \( y = \log_{3}x \) can undergo various transformations, which will enable us to alter its graph easily. Transformations typically fall into categories such as translations (shifts), reflections, and scaling.

Understanding these transformations can help us predict how a graph will look after applying certain mathematical operations.

• **Translations:** These involve shifting the entire graph. For example, adding a constant value outside the function leads to a vertical shift.
• **Reflections:** These happen when you multiply the function by a negative number, flipping it across an axis.
• **Scaling:** Although not covered in this exercise, involves multiplying inside the function which stretches or compresses the graph horizontally.

The exercise focuses on shifting and reflecting the graph, which we will discuss in more detail in other sections.

Change-of-Base Property

The change-of-base property is a key tool in converting logarithms to a different base, making it easier to evaluate or graph them. It's especially useful when your calculator only supports certain bases (like 10 or \( e \)). The change-of-base formula is:

\[ \log_{b}x = \frac{\log_{k}x}{\log_{k}b} \]

This formula enables you to compute \( \log_{3}x \) using common logarithms (base 10) or natural logarithms (base \( e \)). This exercise uses the property to help graph \( y = \log_{3}x \) utilizing graphing utilities that handle base changes internally. Understanding how to apply this property gives you the flexibility to work with logarithmic functions in varying scenarios.

Vertical and Horizontal Shifts

In graphing, shifting involves moving the graph without altering its shape. Vertical and horizontal shifts of a logarithmic function like \( y = \log_{3}x \) are straightforward once you identify the changes in the equation.

**Vertical Shifts** occur when you add or subtract a constant outside the function. The function \( y = 2 + \log_{3}x \) is a classic example where the base graph is moved 2 units upward. The value added determines the direction and magnitude of the shift.

**Horizontal Shifts** occur when you modify the variable inside the function. For instance, in \( y = \log_{3}(x + 2)\), the graph of \( y = \log_{3}x \) shifts 2 units to the left. The shift's direction is opposite to the sign inside the logarithm: plus moves left, minus moves right.

• Vertical shifts affect the height of the graph but not the horizontal position of points.
• Horizontal shifts affect the left or right position, not the vertical position.

These transformations shift the base graph while maintaining its overall shape, crucial for understanding more complex functions.

Graphing Utilities

Graphing utilities are powerful tools that provide a visual perspective on mathematical functions, making understanding of graphs accessible even to beginners. Students often use graphing calculators or software to plot functions like \( y = \log_{3}x \) quickly and accurately.

Graphing utilities come with several benefits:

• **Ease of Use:** They simplify the plotting process, allowing you to visualize functions without manual calculations.
• **Accuracy:** These tools help ensure the graphs are plotted precisely, showing asymptotes and correct curves.
• **Exploration:** You can easily apply transformations and instantly see their effects on a graph.

In this exercise, using a graphing utility allowed students to employ the change-of-base property effortlessly, enabling seamless graphing of functions in different logarithmic bases. It showcases how different manipulations like shifts can be visualized, enhancing comprehension of logarithmic functions.