Problem 12 - 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

Answer: The transformations and their effects are as follows: 1) \(y= \ln x\): No transformation, the basic logarithmic function. 2) \(y= \ln (-x)\): Reflection over the y-axis, which makes the domain negative real numbers. 3) \(y=-\ln (x+3)\): A vertical reflection and a horizontal translation of 3 units to the left. The graph starts from the point (-3,0) and decreases as the x-coordinate increases.

Step 1: Understand the basic properties of logarithmic functions

A logarithmic function has the form \(f(x)=\ln x\), where the logarithm is a transformation of the exponential function \(g(x)=e^x\). The domain of the function \(f(x)\) is all positive real numbers \(x>0\) and the range is all real numbers \((-\infty, \infty)\). The graph of \(f(x)\) is symmetric with respect to the tangent line \(y=x\), which means it passes through the point \((1,0)\) and its slope is 1.

Step 2: Analyze each logarithmic function

(1) \(y=\ln x\): This is the basic logarithmic function without any transformation. (2) \(y=\ln (-x)\): Comparing it to the basic logarithmic function, this graph has a reflection over the y-axis because of the negative sign inside the logarithm. The domain changes to negative real numbers \(x<0\). (3) \(y=-\ln (x+3)\): There are two transformations here: a vertical reflection due to the negative sign in front of the logarithm, and a horizontal translation of 3 units to the left because of the "+3" inside the logarithm.

Step 3: Sketch the logarithmic functions on the same set of axes

(1) \(y=\ln x\): start by drawing the basic logarithmic function, which starts from the origin, passes through the point (1,0), and increases as the x-coordinate increases. (2) \(y=\ln (-x)\): reflect the basic logarithmic function from step 1 over the y-axis, which means mirroring the graph on the left side of the y-axis. The graph starts from minus infinity, passes through the point (-1, 0), and approaches the y-axis as the x-coordinate approaches 0. (3) \(y=-\ln (x+3)\): take the original logarithmic function from step 1 and move it 3 units to the left. This moves the graph's initial point to (-3,0) and its (0,0) point to (-3, -\ln 3). Then, reflect the graph vertically by multiplying the y-coordinates by -1. This will cause the graph to start from the point (-3,0) and decrease as the x-coordinate increases. Put all three graphs on the same set of axes, keeping in mind their respective domains and translations.