Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=\log _{1 / 5} x, g(x)=-\log _{1 / 5}(x-7)\)

Short answer

The function \(g(x) = -\log_{1/5}(x-7)\) is derived from the original function \(f(x) = \log_{1/5}x\) by a horizontal shift of 7 units to the right and a reflection over the x-axis. When graphed, \(f(x)\) shows a descending curve starting at \(x=1, y=0\) and never touching x-axis with asymptote at x=0, while \(g(x)\) shows an ascending curve starting from \(x=7, y=0\) and never touching x-axis with asymptote at x=7.

Step-by-step solution

Identify the transformation

The transformation from \(f(x) = \log_{1/5}x\) to \(g(x) = -\log_{1/5}(x-7)\) consists of two types of transformations. First one is a horizontal shift (translation) of 7 units to the right, represented by the change from \(x\) to \(x-7\) inside the logarithm. The negative sign before the logarithm indicates a reflection over the x-axis. There is no vertical shift or dilation, since the terms outside the logarithm are unchanged apart from the negative sign.

Graph the original function \(f(x)\)

To graph the original function \(f(x) = \log_{1/5}x\), it is crucial to remember that the graph of any logarithmic function will have a vertical asymptote at x=0. The base of the logarithm determines the shape and orientation of the curve. Since \(1/5\) is less than 1, the function is decreasing, indicating the graph should be descending as x moves to the right. The point \(x=1, y=0\) always lies on the graph of the function, since any number to the power of 0 equals 1. So, the graph will start at this point and descend as x increases, getting closer to the x-axis but never touching it.

Graph the transformed function \(g(x)\)

To graph the transformed function \(g(x) = -\log_{1/5}(x-7)\), start with the graph of the original function \(f(x)\), then perform the transformations identified earlier. Shift the graph 7 units to the right, making the vertical asymptote move from x=0 to x=7. Also, reflect the graph over the x-axis - this will invert the descending trend of \(f(x)\), making \(g(x)\) an increasing function. The turning point of \(g(x)\) will be at \(x=7, y=0\), which is 7 units to the right compared to the turning point of \(f(x)\) at \(x=1, y=0\). The graph of \(g(x)\) will start from this point and increase as x moves to the right, getting closer and closer to the x-axis but never touching it.

Key concepts

Function Transformation

A function transformation involves changing the appearance or position of a parent function to create a new function. When working with transformations of logarithmic functions, your goal is to alter the basic form to achieve the desired function. The transformations can include translations, reflections, stretches, and compressions.

• **Translation**: Moving the function's graph horizontally or vertically without altering its shape. In the exercise, we deal with a horizontal translation.
• **Reflection**: Flipping the graph over a specified axis. This exercise includes reflection across the x-axis.
• **Stretching/Compressing**: Changing the size of the graph horizontally or vertically can also occur but is not present in our exercise.

Understand that the transformations are applied in a specific order. Typically, you follow these instructions: first perform horizontal or vertical shifts, followed by reflections, and finally, dilations, if any. For the given problem, we focus on the horizontal shift and reflection.

Horizontal Shift

Horizontal shifts in a function move the graph left or right. This occurs due to changes inside the function's operational components. For example, replacing every **x** with **x - c** in the function notation results in a shift.In the exercise, we shift the function from \(\log_{1/5}{x}\) to \(\log_{1/5}{(x-7)}\). This indicates a translation of the graph 7 units to the right. This transformation affects every x-coordinate of the points on the original graph, pushing them all towards larger x-values by 7 units.

• **Identify the Shift**: Examine the argument of the logarithm. If it has a subtraction like \((x-7)\), it's a right shift by that number of units.
• **Visualize the Changes**: Imagine or draw how the vertical asymptote of the function moves from x = 0 to x = 7.

Doing horizontal shifts correctly ensures that the graph maintains its overall shape but changes its position on the coordinate plane.

Reflection across the x-axis

Reflection is another major transformation which we can perform on functions. Reflecting a graph across the x-axis involves flipping it upside down.For the logarithmic function \(f(x) = \log_{1/5}{x}\), a negative sign on the function like \(-\log_{1/5}{x}\) results in this transformation.

• **Flipping the Graph**: The negative sign causes all positive y-values of the original function to become negative and vice versa.
• **Impact on the Curve**: Since \(f(x)\) is a decreasing function with a vertical asymptote, reflecting it across the x-axis makes the original decreasing pattern rise as it approaches the asymptote.

When applied to the problem's function, the reflection converts the initial downward trend into an upward growth as x increases, starting at the adjusted turning point of the graph.