Problem 47 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of the function \(h(x) = \frac{1}{x} + 2\) is a hyperbola that has been shifted upward by two units. The graph now approaches the line defined by y=2 instead of the x-axis (y=0).

Step 1: Identify the basic function and transformation

Firstly, identify the basic function and the transformation applied to it. In this case, the basic function is \(f(x) = \frac{1}{x}\) and the transformation is `+' (plus) which indicates a vertical shift (upwards)

Step 2: Graph the basic function

Next, draw the graph of the basic function \(f(x) = \frac{1}{x}\). This graph is a hyperbola that approaches but never reaches the x-axis and y-axis.

Step 3: Apply the transformation

Now, apply the transformation. As explained, the '+2' leads to a vertical shift upwards by 2 units which means every point on the graph of \(f(x) = \frac{1}{x}\) is moved two units upwards to obtain the graph for \(h(x) = \frac{1}{x} + 2\).

Step 4: Finalize the graph

Finally, draw the transformed graph by moving every point on the initial graph 2 units upwards. The final graph representing \(h(x) = \frac{1}{x} + 2\) will also be a hyperbola that has been shifted upward by two units. The graph now approaches the line y=2 instead of the x-axis (y=0).

Key Concepts

Transformations of Functions

Transformations of functions are changes that affect the position and shape of a graph without altering its original characteristics. Essentially, these transformations allow you to translate, stretch, compress, or reflect graphs in various ways. Understanding transformations is vital because it gives you the power to predict and modify the behavior of functions easily. When dealing with graphs, two primary transformation types are:

• Vertical and Horizontal Shifts: These move the graph up, down, left, or right.
• Stretches and Compressions: These make the graph taller, shorter, or wider.

In the given exercise, the focus is on a vertical shift transformation that moves the graph upward by adding a constant value to the function.

Vertical Shift

A vertical shift in the graph of a function occurs when you add or subtract a constant from the function. This action moves the graph up or down on the y-axis.For example, if you have a function such as:\[ h(x) = \frac{1}{x} + 2 \]This indicates a vertical shift. The basic function \( f(x) = \frac{1}{x} \) is shifted upwards by 2 units because of the '+2'.Key points to remember about vertical shifts:

• Adding a positive constant moves the graph up.
• Subtracting a constant moves the graph down.

This kind of transformation does not affect the shape of the graph and only changes the position of its horizontal asymptote.

Hyperbola

A hyperbola is the shape formed by the graph of functions like \( f(x) = \frac{1}{x} \). It consists of two separate curves called branches.Characteristics of a hyperbolic graph include:

• Asymptotes: These are lines that the graph approaches but never touches. For \( \frac{1}{x} \), the x-axis \((y=0)\) and y-axis \((x=0)\) are asymptotes.
• Symmetry: Hyperbolas are symmetric about the x-axis and y-axis.

The hyperbola in the exercise is altered by the transformation, effectively shifting its position higher on the graph, but maintaining its overall form and asymptotic properties.

Rational Functions

Rational functions are fractions where both the numerator and the denominator are polynomials. They are a fundamental type of function that often leads to graphs with interesting asymptotic behaviors.In the context of the exercise, the function given is a rational function:\[ h(x) = \frac{1}{x} + 2 \]Rational functions can display many forms, such as linear, quadratic, or as here, inverse variations (hyperbola). Key features include:

• Domain: Values of x that do not make the denominator zero.
• Asymptotes: Horizontal asymptotes are determined by comparing the degree of the numerator and denominator.

By understanding how to transform these functions, you can easily predict changes in their graphs, which is a crucial skill in plotting and interpreting mathematical relationships.