Problem 49 - A Graphical Approach to College Algebra

Question

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=\log _{2}(-x)+1$$

Step-by-Step Solution

Answer

Reflect \( y = \log_{2} x \) across the y-axis, then shift up by 1 unit.

Step 1: Recognize the Base Function

Start by identifying the base function, which is given as \( y = \log_{2} x \). This is the logarithmic function with base 2, which is defined only for positive values of \( x \).

Step 2: Reflect Across the y-axis

The given function is \( y = \log_{2}(-x) + 1 \). Here, \( \log_{2}(-x) \) indicates a reflection of the base function \( y = \log_{2} x \) across the y-axis. This is because substituting \( -x \) in place of \( x \) inverts the graph horizontally.

Step 3: Translate Vertically

The expression \( +1 \) in \( y = \log_{2}(-x) + 1 \) translates the graph of \( y = \log_{2}(-x) \) upwards by 1 unit along the y-axis.

Step 4: Graph the Function

To graph \( y = \log_{2}(-x) + 1 \), start by sketching \( y = \log_{2}(-x) \), which is the reflection of \( y = \log_{2}x \) over the y-axis, resulting in the graph inhabiting the negative x-domain. Then, shift this entire graph up by 1 unit to account for the \(+1\) translation.

Key Concepts

Reflection of Graph

When dealing with transformations of logarithmic functions, a reflection is one of the key changes that can occur. A reflection across the y-axis involves changing the input of the function from \( x \) to \( -x \). This change flips the graph horizontally.

• For our function \( y = \log_2(-x) \), the reflection causes the graph originally on the right side of the y-axis to flip over to the left.
• The domain changes to negative values, meaning instead of being defined for \( x > 0 \), it's now defined for \( x < 0 \).

This type of inversion is common in functions and isn’t limited to logarithms. Anytime you see \( -x \) in place of \( x \), expect the graph to perform this horizontal flip.

Vertical Translation

After reflecting a function, you might also need to perform a vertical translation. This is a vertical shift, moving the graph up or down along the y-axis.

• In our function \( y = \log_2(-x) + 1 \), the '+1' indicates that after reflecting, the graph should be translated upwards by 1 unit.
• This kind of translation adjusts the entire graph equally upwards, affecting every point on the graph.

Vertical translations are often adjustments to a function's base form, allowing it to model data more accurately or fit a desired position on the graph.

Base Function

In any transformation problem, recognizing the base function is crucial. Here, our base function is \( y = \log_2 x \). This is a standard logarithmic function where the input \( x \) must be positive.

• The base of the logarithm, which is 2 in this case, affects the rate and shape of the logarithmic curve.
• Understanding the base function allows you to predict how transformations like vertical shifts or reflections will affect the graph.

It serves as the starting point, or "blueprint," for any graphing transformations.

Graph Transformation

Graph transformations involve systematically altering a graph by shifts, reflections, stretches, or compressions. Each transformation corresponds to a change in the function’s equation.

• The reflection across the y-axis, translating \( x \) to \( -x \), is a horizontal transformation.
• Adding 1 to the function translates the graph vertically, raising it by 1 unit.

Understanding these principles is key to graphing transformations. By following this step-by-step approach, you gain clarity on how the graphical representation of a function changes with its mathematical equation.