Problem 54 - Intermediate Algebra

Question

Graph \(f(x)=\log _{5} x\). Now predict the graphs for \(f(x)=\) \(2 \log _{5} x, f(x)=-4 \log _{5} x\), and \(f(x)=\log _{5}(x+4)\). Graph these three functions on the same set of axes with \(f(x)=\log _{5} x\).

Step-by-Step Solution

Answer

The graphs include a vertical stretch, reflection and stretch, and a horizontal shift.

Step 1: Understanding the Base Graph

The base graph is given by the function \( f(x) = \log_{5}{x} \). This is a logarithmic function with base 5. The graph of a logarithmic function passes through the point \( (1,0) \) because \( \log_{5}{1} = 0 \), and it is approaching negative infinity as x approaches 0 from the right.

Step 2: Graph Transformation 1: Vertical Stretch

The function \( f(x) = 2 \log_{5}{x} \) is a vertical stretch of the original graph. This means that every \( y \)-value for a given \( x \) will be multiplied by 2. If the base function was passing through \( (5, 1) \), now it will pass through \( (5, 2) \) because \( 2\log_{5}{5} = 2 \times 1 = 2 \).

Step 3: Graph Transformation 2: Reflection and Vertical Stretch

The function \( f(x) = -4 \log_{5}{x} \) involves two transformations: a reflection across the x-axis and a vertical stretch by a factor of 4. This changes the value at every point such that \( y \) is multiplied by -4. Thus, for a point \( (5, 1) \) in the base graph, it becomes \( (5, -4) \).

Step 4: Graph Transformation 3: Horizontal Shift

The function \( f(x) = \log_{5}{(x+4)} \) is a horizontal shift of the base graph \( \log_{5}{x} \) to the left by 4 units. So, the point \( (1, 0) \) on the base graph moves to \( (-3, 0) \), consistent with the equation \( x + 4 = 1 \) yielding \( x = -3 \).

Step 5: Plotting the Graphs on the Same Axes

To plot all transformations, start with the basic graph of \( \log_{5}{x} \), then plot each transformation: for \( 2\log_{5}{x} \), double the y-values; for \(-4\log_{5}{x} \) reflect and multiply by 4; and for \(\log_{5}{(x+4)}\) shift back by 4 units horizontally. These transformations should be visually represented on the same set of axes for comparison.

Key Concepts

Function Transformations

Function transformations are modifications to the input or output of a function that change the appearance of its graph. When you apply these transformations to a basic function, it alters how the graph looks without changing its fundamental nature.

In our context of logarithmic functions, transformations make it possible to move, stretch, compress, or flip the graph in various directions.

Types of transformations include:

• Vertical stretches or compressions
• Reflections
• Horizontal or vertical shifts

Understanding these transformations allows you to predict and graph functions without plotting numerous points individually.

Vertical Stretch

A vertical stretch impacts the y-values of a function, making them larger by a specific factor. Imagine pulling the graph away from the x-axis without affecting its horizontal position.

For example, given the function:

• The vertical stretch in the function forms a new graph: if we have the base logarithm function \(f(x) = log_{5}{x}\), and a transformed function \(g(x) = 2 log_{5}{x}\), each y-value is multiplied by 2.

Imagine a point (5,1) on the graph of the base function. By applying a vertical stretch of factor 2, it becomes (5,2). The x-values remain unchanged, but the graph appears taller, stretching upwards.

Reflection

A reflection transformation flips the graph of a function over a specified axis. In our case, we focus on reflection over the x-axis. This transformation affects the y-values by multiplying them by -1, which inverts them vertically.

Take, for instance, the function

• The transformation in \(f(x) = -4 \log_{5}{x}\) involves reflection and vertical stretching.

Here, you multiply each y-value by -4. This results in a flipped and vertically stretched graph. For the base point (5, 1), after transformation, you now have (5, -4). Notice how the graph not only flips down but also stretches compared to the original.

Horizontal Shift

A horizontal shift modifies the graph's position along the x-axis. It doesn’t change the graph’s shape, just its horizontal position. In the exercise, the function \(f(x) = \log_{5}{(x+4)}\) showcases this transformation.

Here’s what happens:

• Adding +4 inside the log function results in shifting the graph to the left by 4 units.

For example, the point (1, 0) on the original graph \( \log_{5}{x} \) shifts to (-3, 0). The graph physically moves left, making it appear to start its increase from a point four units before where it originally began. This horizontal movement doesn't affect the function's vertical nature. It simply drags every point along the x-axis by 4 units.