Problem 53 - College Algebra

Question

In Exercises \(45-56,\) use transformations of \(f(x)-\frac{1}{x}\) or \(f(x)-\frac{1}{x^{2}}\) to graph each rational function. $$h(x)=\frac{1}{x^{2}}-4$$

Step-by-Step Solution

Answer

The graph of function \(h(x)=\frac{1}{x^{2}}-4\) is the curve of the function \(f(x)=\frac{1}{x^{2}}\) shifted downwards by 4 units. The vertical asymptote is at x=0.

Step 1: Identifying the Base function and Transformation

Let's denote the given function, \(h(x)=\frac{1}{x^{2}}-4\) as a transformation of the base function \(f(x)=\frac{1}{x^{2}}\). Here, negative 4 in \(h(x)\) is vertical shifting of the base function by 4 units downwards.

Step 2: Creating the Table of Values

To graph \(h(x)=\frac{1}{x^{2}}-4\), we need to create a table of x-values and corresponding y-values. If we plug in the x-values as -2, -1, 0, 1 and 2. we get the corresponding y-values as -3, -3, -4, -3, -3 respectively.

Step 3: Graphing the Function

The final step is to plot the above points on the graph. Start by graphing the parent function \(f(x) = \frac{1}{x^{2}}\), it's a curve opening upwards with a vertical asymptote at x=0. Now shift this graph 4 units down to graph \(h(x)=\frac{1}{x^{2}}-4\). The vertical asymptote remains the same at x=0.

Key Concepts

Transformations of Functions

Transformations of functions allow us to easily modify and adapt basic graphs to create more complex functions. Understanding transformations is key to working efficiently with different types of functions. There are several types of transformations including:

• Translation: This involves shifting the graph horizontally or vertically without changing its shape.
• Reflection: Here the graph is flipped across a specific axis.
• Stretching/Compressing: This changes the width or height of the graph.

For the function \( h(x) = \frac{1}{x^2} - 4 \), this is a vertical translation of the base function \( f(x) = \frac{1}{x^2} \). Here, the graph of \( f(x) \) is shifted downward by 4 units. Transformations change the way a function behaves without altering its fundamental characteristics. Mastering these adjustments can make graphing much more intuitive and less time-consuming.

Graphing Rational Functions

Rational functions are graphs of equations written in the form \( \frac{P(x)}{Q(x)} \), where \( P \) and \( Q \) are polynomials. The graph of \( f(x) = \frac{1}{x^2} \) has a unique shape characterized by a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \). When graphing rational functions, it's crucial to:

• Identify vertical asymptotes by setting the denominator equal to zero and solving for \( x \).
• Determine horizontal asymptotes by comparing the degrees of the numerator and denominator polynomials.
• Look for any holes in the graph, which occur at zeroes of common factors in \( P \) and \( Q \).

In our example, the function is \( h(x) = \frac{1}{x^2} - 4 \). Since any rational graph's shape is dictated by these asymptotes, understanding their placement and effect helps in accurately drawing the function.

Vertical Shifts

Vertical shifts are a form of translation that moves the graph up or down the coordinate plane. This occurs without affecting the overall shape of the graph. To perform a vertical shift:

• If a constant \( k \) is added, \( f(x) + k \), the graph shifts up by \( k \) units.
• If it is subtracted, \( f(x) - k \), the graph moves down by \( k \) units.

In the given function \( h(x) = \frac{1}{x^2} - 4 \), we perform a vertical shift downward by 4 units. The original graph of \( f(x) = \frac{1}{x^2} \) opens upwards with a vertical asymptote at \( x = 0 \) and moves down uniformly. Despite the shift, the asymptote remains unchanged. Vertical shifts do not alter the domain or asymptotes but only the range of the function, making them an important tool for aligning functions along the y-axis.