Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Explain how the graph of \(f\) can be obtained from the graph of \(y=\frac{1}{x}\) or \(y=\frac{1}{x^{2}} .\) Draw a sketch of the graph of \(f\) by hand. Then generate an accurate depiction of the graph with a graphing calculator. Finally, give the domain and range. $$f(x)=-\frac{0.5}{x^{2}}$$

Step-by-Step Solution

Answer

Reflect \(y=\frac{1}{x^2}\) over the x-axis and vertically shrink by 0.5. Domain: \((-\infty, 0) \cup (0, \infty)\). Range: \((-\infty, 0)\).

Step 1: Identify the Base Function

The given function is related to the base function \( y = \frac{1}{x^2} \). This is because the denominator involves \(x^2\).

Step 2: Identify Transformations

The function \( f(x) = -\frac{0.5}{x^2} \) indicates two transformations applied to \( y = \frac{1}{x^2} \).1. **Vertical Scaling by 0.5:** The factor of 0.5 is a vertical shrink compared to \( y = \frac{1}{x^2} \).2. **Reflection Across the X-axis:** The negative sign indicates a reflection of the graph of \( y = \frac{1}{x^2} \) across the x-axis.

Step 3: Sketch the Function by Hand

Start with the graph of \( y = \frac{1}{x^2} \), which is an upward-opening curve in both the first and the second quadrants.1. Reflect it across the x-axis: Now, the graph will be in the third and fourth quadrants and opens downward.2. Apply the vertical shrink: Instead of passing through points like \((1, 1)\), it will pass through \((1, -0.5)\). The shape remains the same but is closer to the x-axis.

Step 4: Use Graphing Calculator

Input the function \( f(x) = -\frac{0.5}{x^2} \) into the graphing calculator. Ensure the calculator performs the reflection and vertical shrink to visualize the accurate transformation. Compare this depiction with your hand sketch to check for consistency.

Step 5: Determine the Domain

For \( f(x) = -\frac{0.5}{x^2} \), the denominator \(x^2\) cannot be zero, hence \( x eq 0 \). The domain is all real numbers except zero: \( \text{Domain: } (-\infty, 0) \cup (0, \infty) \).

Step 6: Determine the Range

The function implies negative values only since it is reflected over the x-axis. Therefore, the range is all negative real numbers: \( \text{Range: } (-\infty, 0) \).

Key Concepts

Vertical Scaling

Vertical scaling in graph transformations involves stretching or compressing the graph along the y-axis. This transformation changes the height of the graph but keeps the x-coordinates the same.

When you see a factor in front of the function, like the 0.5 in the function \( f(x) = -\frac{0.5}{x^2} \), this indicates a vertical scaling. Specifically, the graph of \( y = \frac{1}{x^2} \) experiences a vertical shrink because 0.5 is less than 1.

• Vertical Shrink: If the factor is between 0 and 1, the graph is compressed towards the x-axis.
• Vertical Stretch: If the factor is greater than 1, the graph is stretched away from the x-axis.

In this function, each point on the graph will be half as far from the x-axis compared to the standard \( y = \frac{1}{x^2} \). This means that points that were once higher will be closer to the axis, maintaining the same x-coordinates.

Reflection Across the X-axis

Reflection across the x-axis is a type of transformation where the graph is flipped over the x-axis. This makes every positive y-value become negative and vice versa.

In the function \( f(x) = -\frac{0.5}{x^2} \), the negative sign indicates that the reflection will occur. You start with the base graph of \( y = \frac{1}{x^2} \), which is positive. Then you reflect it.

• Positive curve: Becomes negative after reflection.
• Flipped orientation: Moves from first and second quadrants to third and fourth quadrants post-reflection.

Reflection changes the orientation of the graph, which can significantly affect its appearance. After reflecting, what was upward opening will now open downward. In this way, reflection across the x-axis also plays a critical role in determining the range of the function.

Domain and Range

The domain and range of a function describe the set of possible input values (domain) and the set of possible output values (range). Understanding these concepts is key to fully grasping a function's behavior.

For the function \( f(x) = -\frac{0.5}{x^2} \):

• Domain: Since the function involves \( x^2 \) in the denominator, \( x \) cannot be zero because division by zero is undefined. Therefore, the domain is all real numbers except zero, written as \( (-\infty, 0) \cup (0, \infty) \).
• Range: After the reflection across the x-axis discussed above, all outputs are negative or less than zero. Hence, the range is all negative real numbers, or \( (-\infty, 0) \).

These properties help us visualize where the graph will lie on the coordinate plane and which values \( f(x) \) can assume.