Tamar plans to bike 13 miles.

Construct a table of values for t\left(r\right), the amount of time it takes Tamar to bike if she rides at a rate of 1 \text{ mph}, 8 \text{ mph}, 10 \text{ mph}, and 15 \text{ mph}.

Graph the function t\left(r\right).

What happens to the function as the values of r \to 0?

What happens to the function as the values of r \to \infty?

The equation for a rational parent function is given by

\displaystyle y=\dfrac{1}{x}

\bm{x}

independent variable

\bm{y}

dependent variable

As the value of one variable increases, the value of the other will decrease.

A **reciprocal function** is a rational function that has a constant numerator. The parent reciprocal function is f\left(x\right) = \dfrac{1}{x}, and is shown on the graph.

Notice that the expression \dfrac{1}{x} is undefined if x=0. Therefore, its domain is \left(-\infty, 0\right) \cup \left(0, \infty\right), which does not include x=0.

Also notice that there is no real value of x that could be substituted into the equation to create f\left(x\right)=0, because 1 divided by any number will never result in 0. So, its range is \left(-\infty, 0\right) \cup \left(0, \infty\right), which does not include y=0 as the function values never reach y=0.

For these reasons, the function f\left(x\right) = \dfrac{1}{x} has two asymptotes: a **vertical asymptote** of x = 0 (the y-axis), and a **horizontal asymptote** of y = 0 (the x-axis). The parent reciprocal function has no x- or y-intercepts, due to its asymptotes.

Examine the end behavior of the reciprocal function as the domain approaches the undefined value of x=0. As x approaches 0 from the negative side, x \to 0^{-}, f\left(x\right) approaches - \infty. As x approaches 0 from the positive side, x \to 0^{+}, f\left(x\right) approaches \infty.

Similarly, when we examine end behavior of the reciprocal function as the domain values approach positive or negative infinity, we see that f\left(x\right) approaches zero. That is, as x approaches + \infty, f\left(x\right) approaches 0 from above and as x approaches -\infty, f\left(x\right) approaches 0 from below.

The equation for another rational parent function is given by

\displaystyle y=\dfrac{1}{x^{2}}

\bm{x}

independent variable

\bm{y}

dependent variable

As the value of one variable increases, the value of the other will decrease.

The graph of this function is shown:

The function f\left(x\right) = \dfrac{1}{x^{2}} has the same two asymptotes as f\left(x\right) = \dfrac{1}{x}: a **vertical asymptote** of x = 0 (the y-axis), and a **horizontal asymptote** of y = 0 (the x-axis).

However, the **branches** (or pieces) of the function are in different quadrants. This causes f\left(x\right) = \dfrac{1}{x^{2}} to have a range of \left(0, \infty \right).

Consider the graphs of the functions f\left(x\right) and g\left(x\right).

a

Identify the function family to which each function belongs.

Worked Solution

b

Compare the asymptotes of f\left(x\right) and g\left(x\right).

Worked Solution

c

Describe the end behavior of each function.

Worked Solution

The relationship between the current, C, (in amperes) and resistance, R, (in ohms) in an electrical circuit is given by: C\left(R\right) = \dfrac {200}{R}where the voltage provided to the circuit is 200\text{ V}.

a

Complete the table.

R | 5 | 10 | 20 | 25 | 40 |
---|---|---|---|---|---|

C\left(R\right) | 8 |

Worked Solution

b

Sketch the relationship between the current and resistance.

Worked Solution

c

Specify any asymptotes and the restrictions on the domain of the function.

Worked Solution

Consider the function, f\left(x\right), shown.

a

Identify the domain and range.

Worked Solution

b

Identify the increasing and decreasing intervals.

Worked Solution

c

Identify the zero(s) of the function.

Worked Solution

Consider the function, f\left(x\right), shown.

a

Identify the increasing and decreasing intervals.

Worked Solution

b

Identify the intercepts of the function.

Worked Solution

c

Describe the end behavior of the function as x\to-\infty and as x\to\infty.

Worked Solution

Idea summary

There are many types of rational functions. Two of the most common parent functions are f\left(x\right)=\dfrac{1}{x} and f\left(x\right)=\dfrac{1}{x^2}.

Drag each slider to change the transformation of the parent function. Check the boxes to change the parent function from f\left(x\right) = \dfrac{1}{x} to f\left(x\right) = \dfrac{1}{x^2}.

How does each slider change the parent function f \left( x \right) = \dfrac{1}{x}?

How does each slider change the parent function f \left( x \right) = \dfrac{1}{x^2}?

What do you notice about the similarities and differences between function with the parents f \left( x \right) = \dfrac{1}{x} and f \left( x \right) = \dfrac{1}{x^2}?

Transformations to the parent function f\left(x\right)=\dfrac{1}{x} create a family of rational functions, given by the following equation:

\displaystyle f\left(x\right)= \dfrac{a}{x-h} + k, h \neq 0

\bm{a}

Stretch \vert a \vert \gt 1 , shrink 0 \lt \vert a \vert \lt 1, reflection across the x-axis when a\lt 0

\bm{h}

Horizontal translation

\bm{k}

Vertical translation, horizontal asymptote at y=k

The same types of transformation can occur for the rational parent function y = \dfrac{1}{x^{2}}.

\displaystyle f\left(x\right)= \dfrac{a}{\left(x-h\right)^{2}} + k, h \neq 0

\bm{a}

Stretch \vert a \vert \gt 1 , shrink 0 \lt \vert a \vert \lt 1, reflection across the x-axis when a\lt 0

\bm{h}

Horizontal translation

\bm{k}

Vertical translation, horizontal asymptote at y=k

Transforming a function affects not only its equation but also its characteristics. Consider these 2 transformations of rational parent functions:

Consider the function g\left(x\right) = \dfrac{1}{x - 1}.

a

What is the transformation of the parent function f\left(x\right) = \dfrac{1}{x}?

Worked Solution

b

Complete the table of values.

x | -1 | 0 | \dfrac{1}{2} | \dfrac{3}{2} | 2 | 3 |
---|---|---|---|---|---|---|

g\left(x\right) |

Worked Solution

c

Sketch a graph of the function.

Worked Solution

Consider the function shown in the graph:

a

Describe the transformation(s) used to get from the graph of y = \dfrac{1}{x} to the graph of this function.

Worked Solution

b

Determine an equation for the function shown in the graph.

Worked Solution

Consider the function y = \dfrac{-5}{\left(x-3\right)^{2}}.

a

Sketch a graph of the function.

Worked Solution

b

What are the equations of the asymptotes of the function?

Worked Solution

c

Using interval notation, what is the domain and range of the function?

Worked Solution

Consider the function f\left(x\right) = \dfrac{3}{x} - 2.

a

Graph f\left(x\right) using transformations of the parent function.

Worked Solution

b

Graph the inverse function f^{-1}\left(x\right).

Worked Solution

Idea summary

Use the parent function to help determine transformations on f\left(x\right) = \dfrac{1}{x}:

\displaystyle f\left(x\right)= \dfrac{a}{x-h} + k, h \neq 0

\bm{a}

Stretch \vert a \vert \gt 1 , shrink 0 \lt \vert a \vert \lt 1, reflection across the x-axis when a \lt 0

\bm{h}

Horizontal translation

\bm{k}

Vertical translation, horizontal asymptote at y=k

A vertical asymptote will occur at x=n where x=n is a zero of the denominator.

The parent function can also be used to help determine transformations on f\left(x\right) = \dfrac{1}{x^{2}}:

\displaystyle f\left(x\right)= \dfrac{a}{\left(x-h\right)^{2}} + k, h \neq 0

\bm{a}

Stretch \vert a \vert \gt 1 , shrink 0 \lt \vert a \vert \lt 1, reflection across the x-axis when a \lt 0

\bm{h}

Horizontal translation

\bm{k}

Vertical translation, horizontal asymptote at y=k

A vertical asymptote will occur at x=n where x=n is a zero of the denominator.