Problem 55 - 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. $$h(x)=\frac{1}{(x-3)^{2}}+1$$

Step-by-Step Solution

Answer

The given function \(h(x)=\frac{1}{(x-3)^{2}}+1\) is a transformation of the basic function \(f(x)=\frac{1}{x^{2}}\) by a horizontal shift of 3 units to the right and a vertical shift of 1 unit upwards. The graph of this function represents a parabola that opens upwards, shifted 3 units to the right and 1 unit upwards, with the minimum point at \((3,1)\).

Step 1: Identify the basic function and transformations

The base function is \(f(x)=\frac{1}{x^{2}}\). The given function \(h(x)=\frac{1}{(x-3)^{2}}+1\) is obtained from the base function by shifting it 3 units to the right and 1 unit upwards. Hence the transformation of the base function includes a horizontal shift and a vertical shift.

Step 2: Perform the horizontal shift

Replace all \(x\)s in \(f(x)\) by \((x-3)\) to perform a horizontal shift of 3 units to the right. This results into a new function \(g(x)=\frac{1}{(x-3)^{2}}\).

Step 3: Perform the vertical shift

Add 1 to \(g(x)\) to shift the graph 1 unit upwards. This gives the final function \(h(x)=\frac{1}{(x-3)^{2}}+1\).

Step 4: Sketch the graph

After the transformations, plot the points on the x-y plane to form the graph of the given function. The graph is a parabola that opens upwards, shifted 3 units to the right and 1 unit upwards. The minimum point is at \((3,1)\), and there is a vertical asymptote at \(x = 3\).

Key Concepts

Transformations of Functions

Transformations of functions involve altering the original function in specific ways to create a new function. These transformations can include translations, reflections, stretches, or compressions. In the example provided, the function transformations focus on translations. Translations in mathematics mean shifting a graph horizontally or vertically without changing its shape.

• The steps involved in transforming \(f(x) = \frac{1}{x^2}\) to \(h(x) = \frac{1}{(x-3)^2} + 1\) comprise a horizontal and a vertical shift.
• This process allows us to start with a familiar function and systematically adjust it to fit different conditions or over different intervals.

Understanding transformations gives insight into how changes in the function's equation impact the final graph's position and orientation on a coordinate plane. It is a fundamental skill to master in graphing rational, polynomial, or any other functions.

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of translations used frequently in graphing transformations. A horizontal shift involves moving a graph along the x-axis, while a vertical shift moves it along the y-axis. In this context:

• The horizontal shift is achieved by replacing \(x\) in the original function with \((x-3)\). This operation shifts the function \(3\) units to the right. Essentially, any input \(x\) now requires an adjustment of \(3\) units, effectively moving the entire graph.
• The vertical shift comes into play by adding \(1\) to the new function. This adjustment elevates every point on the graph upward by 1 unit, repositioning the function along the y-axis.

Both horizontal and vertical shifts do not alter the shape of the graph; instead, they relocate it. By mastering shifts, you can effortlessly adjust the position of any function on a graph, allowing a deeper understanding of how algebraic expressions influence their graphical counterparts.

Graphing Rational Functions

Graphing rational functions involves plotting these functions which are expressed as fractions where both the numerator and the denominator are polynomials. The given function in the exercise, \(h(x) = \frac{1}{(x-3)^2} + 1\), is a rational function derived from its base function \(f(x) = \frac{1}{x^2}\). Here’s a more detailed approach:

• This rational function represents a vertical stretch of the base parabolic graph since it opens upwards.
• After applying the transformations (3 units right, 1 unit up), the vertex moves to the effective minimum point at \((3,1)\).
• There is also a vertical asymptote at \(x = 3\), which indicates where the function is undefined.

When graphing rational functions, you should also pay attention to features such as asymptotes, intercepts, and behavior as \(x\) approaches infinity. This enhances the comprehension required to sketch an accurate representation of the function on the coordinate plane.