Problem 48 - 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. $$y=-4 \log _{2} x-8$$

Step-by-Step Solution

Answer

The transformed function reflects, stretches, and shifts the parent graph.

Step 1: Identify the Parent Function

The parent function is given as \( y = \log_{2} x \). It's a logarithmic function with a base of 2, which has a domain \( x > 0 \) and increases slowly as \( x \) increases.

Step 2: Apply Vertical Stretch

The next transformation involves a vertical stretch. Multiply the parent function by -4. The new step becomes \( y = -4\log_{2}x \). This transformation stretches the graph by a factor of 4 and reflects it across the x-axis.

Step 3: Apply Vertical Shift

Now apply the vertical shift of -8 units. Subtract 8 from the entire function obtained in Step 2. This results in the function \( y = -4\log_{2}x - 8 \). This shifts the entire graph down by 8 units.

Step 4: Graph the Transformed Function

Combine the transformations and graph the function. Start with the known shape of \( y = \log_{2} x \), reflect it across the x-axis to get its negative, stretch it vertically by a factor of 4, and finally shift it downward by 8 units.

Key Concepts

Vertical Stretch

A vertical stretch happens when every point on a graph is moved away from the x-axis, making the graph taller. For logarithmic functions like \( y = \log_{2} x \), if we multiply the function by a positive factor, say 4, every point's distance from the x-axis is multiplied by 4. This makes the slope steeper but retains the overall shape.
Now, when the factor is negative, like -4 in our case, the graph not only stretches by a factor of 4 but also reflects across the x-axis. This reflection essentially flips the graph upside down.
Key points:

• The multiplication factor (in this case -4) determines both stretch and reflection.
• The larger the factor's absolute value, the greater the stretch.
• A negative factor causes reflection over the x-axis.

This transformation leads us from \( y = \log_{2} x \) to \( y = -4\log_{2} x \), drastically altering the graph's appearance.

Vertical Shift

Vertical shifts move every point of a graph up or down by a certain amount. This transformation doesn’t change the shape of the graph, just its vertical position. In our example, we apply a vertical shift of -8.
To shift a graph vertically:

• Add a positive number to shift the graph up.
• Subtract a number to shift the graph down.

For our function, after stretching and reflecting with \( y = -4\log_{2} x \), subtracting 8 gives \( y = -4\log_{2} x - 8 \). This means every point on the graph moves down by 8 units. While the shape remains consistent, the graph's y-values decrease, impacting where it crosses the y-axis and its position relative to the x-axis.

Reflection Across x-axis

Reflection across the x-axis is like flipping a graph over it. This reflection is crucial for our function as it changes positive y-values to negative ones, and vice versa.
To reflect a graph over the x-axis:

• Multiply the entire function by -1.

In the given problem, the step of multiplying by -4 also incorporates reflection because of the negative sign. Starting with \( y = \log_{2} x \), multiplying by -4 reflects the graph so that what was above the x-axis now falls below.
In essence, this not only intensifies (stretches) the graph but completes a visual flip. This transformation affects the graph's direction and orientation, making the negative part more pronounced before applying any vertical shift.