Problem 56 - Algebra and Trigonometry

Question

Use transformations of \(f(x)=\frac{1}{x}\) or \(f(x)=\frac{1}{x^{2}}\) to graph each rational function. $$ h(x)=\frac{1}{(x-3)^{2}}+2 $$

Step-by-Step Solution

Answer

The graph of the function \(h(x)=\frac{1}{(x-3)^{2}}+2\) is a hyperbola centered at the point (3,2) and it is asymptotic to the vertical line \(x=3\) and the horizontal line \(y=2\).

Step 1: Identify the Base Function

Recognize the given function \(h(x)=\frac{1}{(x-3)^{2}}+2\) can be seen as a transformation of the base function \(f(x)=\frac{1}{x^{2}}\). This makes \(f(x)=\frac{1}{x^{2}}\) the base function for this problem.

Step 2: Identify the Transformations

Identify the transformations applied to the base function. In this case, the denominator of the fraction in \(h(x)\) is \((x-3)^2\) instead of \(x^{2}\), which indicates a horizontal shift to the right by 3 units. Also, there is a '+2' at the end, which indicates a vertical upward shift of 2 units.

Step 3: Applying Transformation to the Base Function

Apply these transformations to the base function to obtain the graph of \(h(x)\). Firstly, shift the graph of \(f(x)=\frac{1}{x^{2}}\) three units to the right. Secondly, shift the result two units up. This will result in the graph of \(h(x)\).

Step 4: Sketch the Graph

Sketch the graph of \(h(x)\) based on the transformations. The graph will still be a hyperbola like the base function, but it will be centered at (3,2) instead of the origin. The graph will be asymptotic to the vertical line \(x=3\) and the horizontal line \(y=2\).

Key Concepts

Transformations of Functions

Transformations of functions are all about changing the position, shape, or size of a graph based on certain rules. You can think of the function as a stretchy piece of art that can be moved up, down, left, or right. It might even be stretched or compressed in certain directions.

• Vertical transformations: These involve moving the graph up or down, based on adding or subtracting some value.
• Horizontal transformations: These involve moving the graph left or right, based on adding or subtracting a number inside the function's input.
• Reflections: Sometimes, transformations flip the graph over a line, like the x-axis or y-axis.
• Stretches and compressions: The graph can be stretched taller and thinner or compressed shorter and fatter.

For example, if you have a function like \( f(x) = \frac{1}{x^2} \), you might perform transformations to change how it appears. Instead of being centered at the origin, you may shift it horizontally or vertically. Understanding these transformations helps you predict how a function will behave when changes are made, making graphing more intuitive.

Graphing Rational Functions

Graphing rational functions often involves understanding asymptotes and identifying the basic function shape. A rational function is a ratio of two polynomials. In our example, \( f(x) = \frac{1}{x^2} \), the graph forms a hyperbola.
These functions can feature vertical and horizontal asymptotes:

• Vertical asymptotes occur where the denominator equals zero and the function blows up to infinity.
• Horizontal asymptotes indicate the line that the function approaches as it heads to infinite values of x, but never quite reaches.

Graphing doesn't just involve plotting points. It also involves understanding these asymptotes and shifts. The base shape (a hyperbola) is manipulated by shifting it side to side, up or down, to form the graph of the transformed rational function. Knowing these basics can make graphing rational functions easier and more predictable.

Horizontal and Vertical Shifts

Shifts are a vital transformation type, adjusting where the function sits on the graph. A horizontal shift moves the graph left or right:

• For example, \( f(x - 3) \) shifts the function \( f(x) \) to the right by 3 units.

Whereas, a vertical shift adjusts the graph up or down:

• For example, adding 2 in \( f(x) + 2 \) moves the function upwards by 2 units.

In the problem you encountered, the horizontal shift of 3 units was applied due to the \((x - 3)^2\) in the denominator. The vertical shift of 2 units up came from the addition of 2 outside the fraction.
The result is a graph that relocates the hyperbola center to (3,2) from the origin. By applying these shifts, you're effectively re-centering the asymptotes to where they need to be for the transformed function. Understanding these shifts is key to mastering the graphing of transformed functions.