Question

Describe how the graph of each logarithmic function can be obtained from the graph of \(y=\log _{5} x\). a) \(y=\log _{5}(x-1)+6\) b) \(y=-4 \log _{5} 3 x\) c) \(y=\frac{1}{2} \log _{5}(-x)+7\)

Short answer

a) Right 1, up 6. b) Horizontal compression \( \frac{1}{3} \), vertical stretch 4, reflected over x-axis. c) Reflected over y-axis, vertical compression \( \frac{1}{2} \), up 7.

Step-by-step solution

Identify Base Function

The base function is given as \( y = \log_{5} x \). This is the starting point for all transformations.

Transforming \( y = \log_{5}(x-1)+6 \)

First, note \( x-1 \) moves the graph right by 1 unit. Adding +6 translates the graph up by 6 units. Thus, the transformations are: right 1 and up 6.

Transforming \( y = -4 \log_{5}(3x) \)

The \( 3x \) term inside the logarithm compresses the graph horizontally by a factor of \( \frac{1}{3} \). The \( -4 \) outside the logarithm reflects the graph over the x-axis and stretches it vertically by a factor of 4.

Transforming \( y = \frac{1}{2} \log_{5}(-x)+7 \)

The \( -x \) inside the logarithm reflects the graph over the y-axis. The \( \frac{1}{2} \) outside the logarithm compresses it vertically by a factor of \( \frac{1}{2} \). Finally, adding +7 translates the graph up by 7 units.

Key concepts

graph transformations

Graph transformations are the changes made to the parent function's graph to obtain a new graph. These transformations can include shifts, reflections, stretches, and compressions. The parent function in this case is the logarithmic function, specifically \(y = \log_{5} x\). By altering this function, we can change how the graph looks on the coordinate plane. Whether we move points left or right, up or down, or flip the graph, these adjustments are all part of transforming the graph.

logarithmic functions

Logarithmic functions are the inverse of exponential functions. For the base function \(y = \log_{5} x\), we interpret it as 'the power to which 5 must be raised to produce x.' These functions grow more slowly than linear or exponential functions and have distinctive curved graphs that always pass through the point \( (1, 0) \) if the base is greater than 1. Understanding how various constants affect this graph can help you predict changes and shifts.

horizontal shift

A horizontal shift moves the graph left or right along the x-axis. For our function \(y = \log_{5}(x-1)+6\), the \(x-1\) inside the logarithm translates the graph to the right by 1 unit. To visualize this, consider every point on the graph moving 1 unit in the positive x direction. Horizontal shifts are determined by the value subtracted or added within the function's argument.

vertical shift

Vertical shifts move the graph up or down along the y-axis. In the function \(y = \log_{5}(x-1)+6\), the +6 outside the logarithm moves the graph up by 6 units. Each point on the graph is raised 6 units in the positive y direction. Vertical shifts are easier to recognize as they add or subtract directly to the function's output.

reflection

Reflections flip the graph over a specific axis. In the function \(y = \-4 \log_{5}(3x)\), the negative coefficient, -4, reflects the graph over the x-axis, turning it upside down. Another example is \(y = \frac{1}{2} \log_{5}(-x)+7\), where the \(-x\) inside the logarithm reflects the graph over the y-axis. Reflections change the orientation of the graph while preserving its shape.

compression and stretch

Compression and stretch transformations alter the graph's width or height. In \(y = \-4 \log_{5}(3x)\), the 3 inside the logarithm compresses the graph horizontally by a factor of \(1/3\), while the -4 stretches it vertically by a factor of 4. In \(y = \frac{1}{2} \log_{5}(-x)+7\), \(\frac{1}{2}\) compresses the graph vertically by a factor of \(\frac{1}{2}\). These transformations reshape the graph to be either narrower, wider, taller, or shorter depending on the multiplier used.