Problem 14 - 5 Steps to A 5: AP Calculus AB 2021

Question

On the same set of axes, sketch the graphs of: (1) \(y=\ln x\) (2) \(y=\ln (-x)\) (3) \(y=-\ln (x+3)\)

Step-by-Step Solution

Answer

Based on the given solution, analyze and describe the differences between the graphs of the functions \(y=\ln x\), \(y=\ln(-x)\), and \(y=-\ln(x+3)\). The differences between the graphs of the three functions can be described as follows: 1. \(y=\ln x\) represents the basic natural logarithm curve, which starts from negative infinity and passes through the point (1,0). Its domain is restricted to positive x-values. 2. \(y=\ln(-x)\) is a reflection of the basic natural logarithm curve over the y-axis. This graph passes through the point (-1,0) and extends horizontally only on negative x-values. 3. \(y=-\ln(x+3)\) brings two transformations to the basic natural logarithm curve: horizontal shift 3 units to the left and vertical reflection. This graph passes through the point (2,0) and decreases to negative infinity for x-values greater than -3. In summary, all three functions are transformations of the basic natural logarithm curve, with specific reflection and shifting properties that make each graph distinct from the others.

Step 1: Plotting the basic natural logarithm function

To plot the basic natural logarithm function (\(y=\ln x\)), recall that the domain of this function is \((0, \infty)\). It's also worth noting that as \(x\) approaches zero, \(y\) tends to negative infinity, and as \(x\) increases, \(y\) increases slowly to positive infinity. Thus, you will get a curve which starts from negative infinity, passes through the point \((1,0)\), and goes upward slowly to the right.

Step 2: Plotting the \(y=\ln(-x)\) function

Recall that the domain of natural logarithm function is positive numbers. In the case of \(y=\ln(-x)\), we want to take the natural logarithm of negative numbers. To do this, we need a reflection over the y-axis from the basic function, which means the new domain is \((-\infty, 0)\). This new curve will pass through the point \((-1,0)\) and will be the mirror image of \(y=\ln x\) on the negative side of the x-axis.

Step 3: Plotting the \(y=-\ln(x+3)\) function

The given function brings two transformations to the basic logarithm function: horizontal shift and vertical reflection. The horizontal shift, \((x+3)\), represents a shift of 3 units to the left. The original natural logarithm function will be shifted now, having the new domain of \((-3, \infty)\). Furthermore, the function is vertically reflected due to the negative sign in front of the logarithm. This means that the curve will pass through the point \((2,0)\) and will be a reflection of the basic logarithm curve. As \(x\) approaches \(-3\), the function will tend towards positive infinity, and as \(x\) increases, the function will decrease slowly to reach negative infinity.

Step 4: Putting it all together

On the same set of axes, plot the three functions: - \(y=\ln x\): the basic natural logarithm curve starting from negative infinity, passing through \((1,0)\) and moving upwards slowly for positive x-values. - \(y=\ln(-x)\): a reflection of the basic curve over the y-axis, passing through \((-1,0)\), and extending horizontally only on the negative x-values. - \(y=-\ln(x+3)\): a combination of horizontal shift 3 units to the left and vertical reflection of the basic curve, which passes through \((2,0)\) and decreases to negative infinity for the increasing x-values greater than -3. Once you've plotted all three functions, you should have a clear visual of how the given functions are related and how they differ from each other on the same set of axes.

Key Concepts

Natural Logarithm Properties

Understanding the properties of the natural logarithm is pivotal in grasping its behavior when graphing. The natural logarithm, denoted as \(\ln x\), has a specific set of characteristics that define its operations and interactions with other mathematical entities. One fundamental property is that \(\ln 1 = 0\), as the power to which the base \(e\) must be raised to get 1 is zero. Another key property is the inverse relationship with the exponential function; \(\ln(e^x) = x\) and \(e^{\ln x} = x\).

The natural logarithm also follows various logarithmic laws, such as:

• Product Rule: \(\ln(ab) = \ln a + \ln b\)
• Quotient Rule: \(\ln(\frac{a}{b}) = \ln a - \ln b\)
• Power Rule: \(\ln(a^r) = r \ln a\)

These properties are used when transforming and graphing natural logarithm functions, and they are essential in understanding the solutions for our exercise.

Transformations of Logarithmic Functions

Transformations of logarithmic functions involve altering the basic \(\ln x\) curve through operations like reflection, stretching, compressing, and translating. The graphs sketched in our exercise demonstrate several of these transformations.

For instance, \(y = \ln(-x)\) includes a reflection across the y-axis, altering the original domain to negative x-values. Similarly, \(y = -\ln(x+3)\) shows both a horizontal shift and a vertical reflection. A horizontal shift is realized by adding or subtracting a constant from \(x\) inside the logarithm, moving the graph left or right. Vertical reflection is indicated by a negative coefficient in front of the logarithm, flipping the graph upside down.

Asymptotic Behavior of Log Functions

The asymptotic behavior of logarithmic functions is a crucial concept describing how these functions behave as they approach certain lines or points. A standard natural logarithm function \(y = \ln x\) approaches negative infinity as \(x\) nears zero, which we call a vertical asymptote at \(x=0\). The function also slowly climbs towards positive infinity as \(x\) increases, but it never actually reaches infinity.

In the exercise, \(y = \ln(-x)\) exhibits a vertical asymptote at \(x=0\) but on the negative side of the x-axis due to a reflection. \(y = -\ln(x+3)\) has a vertical asymptote at \(x=-3\), shifted left by 3 units from the y-axis. Understanding these behaviors is necessary to accurately graph natural logarithm functions.

Domain and Range of Logarithmic Functions

The concept of domain and range is vital when graphing any function, including logarithms. The domain refers to the set of all possible input values (x-values) for which the function is defined, while the range is the set of possible output values (y-values).

For a basic natural logarithm function \(y = \ln x\), the domain is \(0, \infty)\) because logarithms are only defined for positive numbers. The range, however, is all real numbers because \(\ln x\) can take any value from negative to positive infinity.

In our exercise, the domain for \(y = \ln(-x)\) is \(\infty, 0)\) due to the reflection across the y-axis. For \(y = -\ln(x+3)\), the domain is altered to \(\infty, -3)\) because of the horizontal shift. Understanding domains and ranges allows for an accurate depiction of these functions graphically.