Problem 50 - A Graphical Approach to College Algebra

Question

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=-\log _{2}(-x)$$

Step-by-Step Solution

Answer

Reflect both across the y-axis and x-axis from \( y = \log_2{x} \), then draw.

Step 1: Understand the Basic Function

The function given is based on the logarithmic function \( y = \log_2{x} \). This is the parent function, which is defined for \( x > 0 \) and has properties such as a vertical asymptote at \( x = 0 \), passing through the point \( (1, 0) \), and increasing with a slope that becomes less steep as \( x \) becomes larger.

Step 2: Reflect across y-axis using -x

To obtain the function \( y = \log_2(-x) \) from \( y = \log_2{x} \), we replace \( x \) with \( -x \). This results in a reflection of the graph of \( y = \log_2{x} \) across the y-axis. Now, the domain is defined for \( x < 0 \), and the graph will never intersect the y-axis (as there is a vertical asymptote at \( x = 0 \)).

Step 3: Reflect across x-axis using -y

The next transformation involves reflecting the function \( y = \log_2(-x) \) across the x-axis. This is achieved by multiplying the whole function by \(-1\), leading to \( y = -\log_2(-x) \). This flips the graph vertically, so any positive y-values become negative and vice versa. The vertical asymptote remains at \( x = 0 \).

Step 4: Graph the Transformed Function

To sketch \( y = -\log_2(-x) \), start by plotting a few key points found from evaluating the function for select x-values (where \( x < 0 \)): For instance, if \( x = -1 \), \( y = -\log_2(1) = 0 \); if \( x = -2 \), \( y = -1 \). The graph will approach the vertical asymptote at \( x = 0 \) from the left, never crossing or touching it, and continue indefinitely downwards as it moves left. It will have a general decreasing shape similar to a flipped and sideways logarithmic curve to the left of the y-axis.

Key Concepts

Graph Reflections

Graph reflections are transformations that flip a graph over an axis. In this case, we perform two reflections: first across the y-axis and then across the x-axis to get the graph of the function from its parent function.

1. **Reflection across the y-axis**: This occurs when we replace every "x" in the function with "-x". For the parent function \( y = \log_2{x} \), this transformation changes it to \( y = \log_2(-x) \). This reflects the graph across the y-axis, effectively flipping it horizontally. The domain of the function now becomes \( x < 0 \), which means the graph exists only on the left side of the y-axis.

2. **Reflection across the x-axis**: To achieve this, we multiply the entire function by -1. For \( y = \log_2(-x) \), reflecting across the x-axis results in \( y = -\log_2(-x) \). This flips the graph vertically. Hence, if any points were above the x-axis, they are now below and vice versa.

Graph reflections are important in understanding how complex graphs are derived from simpler ones by manipulating their basic shapes through these flips.

Vertical Asymptote

A vertical asymptote is a line that a graph approaches but never actually touches or crosses. For logarithmic functions of the form \( y = \log_b{x} \), the vertical asymptote is typically at \( x = 0 \) since the function is undefined for non-positive x-values.

In the case of the transformed function \( y = -\log_2(-x) \), the vertical asymptote remains at \( x = 0 \). This is because the transformation involves a reflection across the y-axis where we have replaced \( x \) with \(-x\), but it does not change the position of the asymptote.

As you sketch the graph, you'll notice that as \( x \) approaches 0 from the negative side, the graph tends to decrease towards negative infinity. This behavior highlights the asymptotic nature of the vertical line at \( x = 0 \). In essence, the vertical asymptote is a key feature in predicting the end behavior of logarithmic functions, especially after transformations that involve reflections.

Logarithmic Function

Logarithmic functions are inverse operations of exponential functions. In their most basic form, they are written as \( y = \log_b{x} \), where \( b \) is the base of the logarithm, commonly a number greater than 1. It indicates the power to which the base must be raised to get x.

The parent function, \( y = \log_2{x} \), is the foundation for understanding more complex logarithmic transformations. This function is defined for positive x-values, with a somewhat steep increase that levels out as x grows larger.

• The logarithmic function passes through the point \( (1, 0) \), because \( \log_2{1} = 0 \).
• It has a vertical asymptote at \( x = 0 \), stopping the graph from defining negative or zero values for x.

In the transformed graph of \( y = -\log_2(-x) \), we observe how this parent function behaves after reflections. These transformations are crucial in understanding shifts in domain and range, as they allow the examination of how negative log values or new critical points can emerge.
Understanding these characteristics helps in graphing logarithmic functions effectively and anticipating their behaviors with different transformations.