Question

Graph the functions \(f(x)=\frac{1}{x^{2}}\) \(g(x)=\frac{-8}{(x+6)^{2}},\) and \(h(x)=\frac{4}{x^{2}-4 x+4}-3\) Discuss the characteristics of the graphs and identify any common features.

Short answer

All functions form hyperbolas with vertical and horizontal asymptotes, sharing a common basic shape despite different shifts and transformations.

Step-by-step solution

- Graph the function f(x)

Graph the function \(f(x) = \frac{1}{x^2}\). This function is a hyperbola that opens upwards. Its vertical asymptote is at \(x = 0\) and the horizontal asymptote is at \(y = 0\). The graph approaches these asymptotes but never touches them.

- Graph the function g(x)

Graph the function \(g(x) = \frac{-8}{(x+6)^2}\). This is also a hyperbola, but it opens downwards. The function is a transformation of \(f(x)\) where it is shifted 6 units to the left and vertically stretched by a factor of 8 with a reflection across the x-axis. Its vertical asymptote is at \(x = -6\) and the horizontal asymptote is at \(y = 0\).

- Graph the function h(x)

Graph the function \(h(x) = \frac{4}{x^2 - 4x + 4} - 3\). Rewrite the denominator as \((x-2)^2\). The function becomes \(\frac{4}{(x-2)^2} - 3\), which is a transformation of \(f(x) = \frac{4}{x^2}\). It shifts 2 units to the right and 3 units down. Its vertical asymptote is at \(x = 2\) and the horizontal asymptote is at \(y = -3\).

- Compare and identify common features

All three functions are hyperbolas with vertical and horizontal asymptotes. Function \(f(x)\) has vertical asymptote at \(x = 0\) and horizontal asymptote at \(y = 0\). Function \(g(x)\) has vertical asymptote at \(x = -6\) and horizontal asymptote at \(y = 0\). Function \(h(x)\) has vertical asymptote at \(x = 2\) and horizontal asymptote at \(y = -3\). Despite shifts and reflections, they share the basic hyperbolic shape.

Key concepts

hyperbolas

A hyperbola is a type of smooth curve lying in a plane. For rational functions, it often appears as two separate curves. These curves occur because the denominator of the function can approach zero from either the positive or negative side, causing the function to approach infinity in opposite directions.

Hyperbolas have some distinct characteristics:

• They have two branches.
• Each branch approaches, but never touches, the asymptotes.
• The center of a hyperbola refers to the point where the asymptotes intersect.

Let's analyze the hyperbolic behavior using the functions from the exercise.

The function \(f(x) = \frac{1}{x^2}\) generates a hyperbola that opens upwards. It has a vertical asymptote at \(x = 0\) and a horizontal asymptote at \(y = 0\). Similarly, \(g(x) = \frac{-8}{(x+6)^2}\) forms a hyperbola that opens downwards, and \(h(x) = \frac{4}{(x-2)^2} - 3\) undergoes several transformations but retains the hyperbolic nature.

vertical asymptotes

Vertical asymptotes occur in functions where the denominator can equal zero, making the function undefined. As we approach these points, the function's value becomes extremely large positive or negative.

They can severely affect graph behavior and will always be found in rational functions.

For the given functions:

• \(f(x) = \frac{1}{x^2}\) has a vertical asymptote at \(x = 0\).
• \(g(x) = \frac{-8}{(x+6)^2}\) shifts the vertical asymptote to \(x = -6\).
• \(h(x) = \frac{4}{(x-2)^2} - 3\) moves the vertical asymptote to \(x = 2\).

Note that the transformations (shifts) in \(g(x)\) and \(h(x)\) result in the vertical asymptotes moving left or right from the original function \(f(x)\).

horizontal asymptotes

Horizontal asymptotes occur in rational functions when the value of the function approaches a fixed value as \(x\) goes to infinity or negative infinity. These asymptotes represent the end behavior of the function.

Here’s how horizontal asymptotes relate to our functions:

• \(f(x) = \frac{1}{x^2}\) has a horizontal asymptote at \(y = 0\) because as \(x\) becomes very large or very small, the function's value approaches zero.
• \(g(x) = \frac{-8}{(x+6)^2}\) also has a horizontal asymptote at \(y = 0\) because it is a transformed version of \(f(x)\) but scaled and shifted.
• \(h(x) = \frac{4}{(x-2)^2} - 3\) has a horizontal asymptote at \(y = -3\). The -3 represents a shift downward by 3 units.

The horizontal asymptote helps to quickly understand the function's behavior at extreme values of \(x\).

function transformations

Function transformations shift or change the graph of a function in various ways.

They can include translations, reflections, dilations, and stretches.

To understand transformations thoroughly, consider these types:

• **Shifts (translations):** Moving the graph left, right, up, or down. Example from the exercise: \(g(x) = \frac{-8}{(x+6)^2}\) involves a left shift by 6 units.
• **Reflections:** Flipping the graph across an axis. In \(g(x)\), the negative sign reflects the graph across the x-axis.
• **Stretches and Dilations:** Scaling the graph by compressing or expanding it. For instance, the factor of 8 in \(g(x)\) vertically stretches it by 8 times.

The exercise also shows these transformations in function \(h(x) = \frac{4}{(x-2)^2} - 3\): a right shift by 2 units and downward shift by 3 units. Recognizing these helps in graphing functions accurately and anticipating their behaviors.