Question

a) Sketch the graph of \(y=\log _{2} x,\) and then apply, in order, each of the following transformations. Reflect in the \(y\) -axis. Translate vertically 5 units up. b) Write the equation of the final transformed image.

Short answer

The final equation after the transformations is \( y = \log_{2} (-x) + 5 \).

Step-by-step solution

- Sketch the Graph of the Basic Function

First, sketch the graph of the basic logarithmic function, \( y = \log_{2} x \). This graph passes through the point (1,0) and has a vertical asymptote at \( x = 0 \). It increases slowly to the right and is always increasing.

- Reflect the Graph in the y-axis

Next, reflect the graph of \( y = \log_{2} x \) in the y-axis. To do this, we replace \( x \) with \( -x \) in the equation, resulting in the new equation \( y = \log_{2} (-x) \). This reflection reverses the direction of the graph, and it now passes through (-1, 0).

- Translate Vertically 5 Units Up

Finally, translate the reflected graph 5 units upwards. To achieve this, add 5 to the existing equation. Thus, the new equation becomes \( y = \log_{2} (-x) + 5 \). Now the transformed graph has the same shape but is shifted vertically upward by 5 units.

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. The basic form of a logarithmic function is given as \( y = \log_b x \), where \( b\) (the base) is a positive real number not equal to 1. This function tells us what power we need to raise \( b\) to get \( x\). Also, these functions have a vertical asymptote at \( x = 0\), meaning the graph approaches this line but never touches or crosses it. For example, with the base 2, \( y = \log_2 x\) passes through the point (1,0) and increases slowly to the right.

Graph Transformations

Graph transformations modify the appearance and position of a graph without changing its overall shape. These transformations include translations, reflections, stretches, and compressions. To visualize this, start with the basic graph of a logarithmic function like \( y = \log_2 x\). This graph can undergo various modifications, such as shifting up, down, left, or right, and reflecting across the axes. When we apply transformations, we alter the equation accordingly.

Vertical Translation

Vertical translation involves moving an entire graph up or down along the y-axis. If you add a positive number to the function, the graph shifts upward. Subtracting a number shifts it downward. For instance, with our graph from before, if we take \( y = \log_2 x\) and we add 5, we get the new function \( y = \log_2 x + 5\). Each point on the original graph moves 5 units higher, effectively raising the entire graph.

Reflection Over y-axis

Reflecting a graph over the y-axis means flipping it horizontally across the vertical axis. To do this, replace \( x\) with \(-x\) in the function's equation. For our original logarithmic function, \( y = \log_2 x\), the reflection would be \( y = \log_2(-x)\). This transformation reverses the direction of the graph. While \( y = \log_2 x\) increases to the right, \( y = \log_2(-x)\) increases to the left and passes through (-1, 0).