Problem 14 - Intermediate Algebra

Question

Fill in the blanks. a. The graph of \(g(x)=4+\log x\) is similar to the graph of \(f(x)=\log x,\) but it is translated 4 units ____. The graph of \(g(x)=\log _{4}(x+2)\) is similar to the graph of \(f(x)=\log _{4} x,\) but it is translated 2 units to the ____.

Step-by-Step Solution

Answer

a. up; left.

Step 1: Understanding Logarithmic Functions

Logarithmic functions are transformations of their basic form. The function \( f(x) = \log x \) is a basic logarithmic function with the base of 10, which can transform by shifting vertically or horizontally.

Step 2: Analyze the Graph of \(g(x)=4+\log x\)

The function \( g(x) = 4 + \log x \) is a vertical shift of \( f(x) = \log x \). The entire graph of \( \log x \) is moved up by 4 units because of the addition of 4. Thus, the graph of \( g(x) \) is translated 4 units up.

Step 3: Analyze the Graph of \(g(x)=\log_{4}(x+2)\)

The function \( g(x) = \log_{4}(x+2) \) is a horizontal shift of the function \( f(x) = \log_{4} x \). The term \( x + 2 \) implies the graph is shifted to the left by 2 units. In general, adding to \( x \) results in a shift in the opposite direction.

Key Concepts

Vertical and Horizontal Shifts

In logarithmic functions, shifts can occur vertically or horizontally, providing flexibility in how these functions can be represented graphically. A vertical shift happens when a constant is added or subtracted from the entire function, while a horizontal shift occurs through changes made specifically to the input variable (inside the function's argument).

For example, in the function \( g(x) = 4 + \log x \), the addition of 4 results in a vertical shift upwards by 4 units. This vertical movement modifies the range, maintaining the same shape and domain but elevating every point of the graph by a uniform 4 units.

Conversely, the function \( g(x) = \log_4(x + 2) \) demonstrates a horizontal shift. By adding 2 inside the logarithm, we take the whole graph of \( f(x) = \log_4 x \) and shift it to the left by 2 units. With horizontal shifts, remember:

• Add within the argument \((x + c)\) shifts left.
• Subtract within the argument \((x - c)\) shifts right.

This concept reflects the inverse nature of such transformations on the horizontal axis, affecting the domain specifically.

Transformations of Functions

Transforming functions is a core aspect of understanding how graphs change and adapt. Logarithmic functions, among others, can be transformed straightforwardly through shifts, stretches, or compressions.

Common transformations include:

• **Vertical Shifts:** Adding or subtracting a constant moves the graph up or down.
• **Horizontal Shifts:** Adjustments to the function's variable, as we saw in \( \log_4(x + 2) \), move the graph left or right.
• **Vertical Stretches/Compressions:** Multiplying the entire function by a constant greater than 1 stretches it, while between 0 and 1 compresses it.
• **Horizontal Stretches/Compressions:** Multiplying the variable inside the function by a constant acts inversely compared to vertical transformations.

Transformations allow us to manipulate basic functions to match more complex real-world behavior. By mastering these transformations, we can transform any base logarithmic function into a form that suits our needs.

Understanding each transformation's effect is crucial for analyzing and graphing functions accurately.

Graph Translations

Graph translations are a vital tool in understanding how functions behave. When we translate a graph, we move it but do not alter its shape or orientation.

Here's a recap of shifts as translations:

• **Vertical Translations:** These occur when we add or subtract a constant outside the function—impacting the y-values and moving the graph up or down.
• **Horizontal Translations:** By adding or subtracting inside the function argument (often inside brackets), we shift the graph left or right without changing the x-axis location otherwise.

Understanding translations involves recognizing how these movements impact the function without distorting its shape.

Visualizing these translations helps in predicting the behavior of the graph under different transformations. They are fundamental in preparing the graph for more complex operations and making connections to real-world applications where such transformations are necessary.

Whether dealing with logarithmic or other functions, mastering the idea of graph translations equips you with the confidence to tackle any new graph-related challenge.