unit 8 homework 6 graphing reciprocal functions

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

3.7e: Exercises for the reciprocal function

• Last updated
• Save as PDF
• Page ID 44351

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

A: Graph translations of \( \dfrac{1}{x} \) 

Exercise \(\PageIndex{A}\) 

\( \bigstar \) Graph the given function. Identify the translations on  \(y = \dfrac{1}{x}\) used to sketch the graph. Then state the domain and range.

\( \bigstar \) Use the transformations to graph the following functions.

\( \bigstar \) Graph using translations of \( \dfrac{1}{x} \) by first using division to rewrite the function.

B: Construct a graph from a verbal description

Exercise \(\PageIndex{B}\)

\( \bigstar \) Use the given transformation to graph the function. Note the vertical and horizontal asymptotes.

• The reciprocal function shifted up two units.
• The reciprocal function shifted down one unit and left three units.
• The reciprocal squared function shifted to the right \(2\) units.
• The reciprocal squared function shifted down \(2\) units and right \(1\) unit.

\( \bigstar \)

Chapter 8: More Functions and Identities

Exercises: 8.3 The Reciprocal Functions Exercises

Practice each skill in the Homework Problems listed.

• Evaluate the reciprocal trig functions for angles in degrees or radians #1–20
• Find values or expressions for the six trig ratios #21–28
• Evaluate the reciprocal trig functions in applications #29–32
• Given one trig ratio, find the others #33–46, 71–80
• Evaluate expressions exactly #47–52
• Graph the secant, cosecant, and cotangent functions #53–58
• Identify graphs of the reciprocal trig functions #59–64
• Solve equations in secant, cosecant, and cotangent #65–70
• Use identities to simplify or evaluate expressions #81–94

Suggested Homework

Exercises for 8.3 The Reciprocal Functions

Exercise group.

For Problems 1–8, evaluate. Round answers to 3 decimal places.

[latex]\csc 27°[/latex]

[latex]\sec 8°[/latex]

[latex]\cot 65°[/latex]

[latex]\csc 11°[/latex]

[latex]\sec 1.4[/latex]

[latex]\cot 4.3[/latex]

[latex]\csc \dfrac{5\pi}{16}[/latex]

[latex]\sec \dfrac{7\pi}{20}[/latex]

For Problems 9–16, evaluate. Give exact values.

[latex]\csc 30°[/latex]

[latex]\sec 0°[/latex]

[latex]\cot 45[/latex]

[latex]\csc 60°[/latex]

[latex]\sec 150°[/latex]

[latex]\cot 120°[/latex]

[latex]\csc 135°[/latex]

[latex]\sec 270°[/latex]

For Problems 17–18, complete the tables with exact values.

Evaluate. Round answers to three decimal places.

• [latex]\displaystyle \cos 0.2[/latex]
• [latex]\displaystyle (\cos 0.2)^{-1}[/latex]
• [latex]\displaystyle \cos^{-1}0.2[/latex]
• [latex]\displaystyle \dfrac{1}{\cos 0.2}[/latex]
• [latex]\displaystyle \cos\dfrac{1}{0.2}[/latex]
• [latex]\displaystyle \sec 0.2[/latex]
• [latex]\displaystyle \tan 3.2[/latex]
• [latex]\displaystyle \tan^{-1} 3.2[/latex]
• [latex]\displaystyle \cot 3.2[/latex]
• [latex]\displaystyle \dfrac{1}{\tan 3.2}[/latex]
• [latex]\displaystyle \tan \dfrac{1}{3.2}[/latex]
• [latex]\displaystyle (\tan 3.2)^{-1}[/latex]

For Problems 21–28, find exact values for the six trigonometric ratios of the angle [latex]\theta{.}[/latex]

• Write an expression for the distance, [latex]d{,}[/latex] that sunlight travels through a layer of atmosphere of thickness [latex]h{.}[/latex]
• Find the distance (to the nearest mile) that sunlight travels through a 100-mile layer of atmosphere when the sun is [latex]40°[/latex] above the horizon.
• Use the figure to write an expression for the radius, [latex]r{,}[/latex] of a curve whose degree of curvature is [latex]\theta{.}[/latex] (Hint: The bisector of the angle [latex]\theta[/latex] is perpendicular to the chord.)
• Find the radius of a curve whose degree of curvature is [latex]43°{.}[/latex]

When a plane is tilted by an angle [latex]\theta[/latex] from the horizontal, the time required for a ball starting from rest to roll a horizontal distance of [latex]l[/latex] feet on the plane is

[latex]t=\sqrt{\dfrac{l}{8}\csc(2\theta)}~~ {seconds}[/latex]

• How long, to the nearest 0.01 second, will it take the ball to roll 2 feet horizontally on a plane tilted by [latex]12°{?}[/latex]
• Solve the formula for [latex]l[/latex] in terms of [latex]t[/latex] and [latex]\theta{.}[/latex]

After a heavy rainfall, the depth, [latex]D{,}[/latex] of the runoff flow at a distance [latex]x[/latex] feet from the watershed down a slope at angle [latex]\alpha[/latex] is given by

[latex]D=(kx)^{0.6}(\cot \alpha)^{0.3}~~{inches}[/latex]

where [latex]k[/latex] is a constant determined by the surface roughness and the intensity of the runoff.

• How deep, to the nearest 0.01 inch, is the runoff 100 feet down a slope of [latex]10°[/latex] if [latex]k=0.0006{?}[/latex]
• Solve the formula for [latex]x[/latex] in terms of [latex]D[/latex] and [latex]\alpha{.}[/latex]

For Problems 33–38, write algebraic expressions for the six trigonometric ratios of the angle [latex]\theta{.}[/latex]

The diagram shows a unit circle. Find six line segments whose lengths are, respectively, [latex]\sin t,~ \cos t,~ \tan t,~ \sec t,~ \csc t,[/latex] and [latex]\cot t{.}[/latex]

Use the figure in Problem 39 to find each area in terms of the angle [latex]t{.}[/latex]

• [latex]\displaystyle \triangle OAC[/latex]
• [latex]\displaystyle \triangle OBD[/latex]
• sector [latex]OAC[/latex]
• [latex]\displaystyle \triangle OFE[/latex]

For Problems 41–46, sketch the reference angle and find exact values for all six trigonometric functions of the angle.

[latex]\sec \theta = 2,~~\theta[/latex] in Quadrant IV

[latex]\csc \phi = 4,~~\phi[/latex] in Quadrant II

[latex]\csc \alpha = 3,~~\alpha[/latex] in Quadrant I

[latex]\sec \beta = 4,~~\beta[/latex] in Quadrant IV

[latex]\cot \gamma = \dfrac{1}{4},~~\gamma[/latex] in Quadrant III

[latex]\tan \theta = 6,~~\theta[/latex] in Quadrant I

For Problems 47–52, evaluate.

[latex]4 \cot \dfrac{\pi}{3} + 2\sec \dfrac{\pi}{4}[/latex]

[latex]\dfrac{1}{2} \csc \dfrac{\pi}{6} - \dfrac{1}{4}\cot \dfrac{\pi}{6}[/latex]

[latex]\dfrac{1}{2} \csc \dfrac{5\pi}{3}\cot \dfrac{3\pi}{4}[/latex]

[latex]6 \cot \dfrac{7\pi}{6} \sec \dfrac{5\pi}{4}[/latex]

[latex]\left(\csc \dfrac{2\pi}{3} - \sec \dfrac{3\pi}{4}\right)^2[/latex]

[latex]\sec^2 \dfrac{5\pi}{6} \csc^2 \dfrac{4\pi}{3}[/latex]

Complete the table and sketch a graph of [latex]y=\sec x{.}[/latex]

Complete the table and sketch a graph of [latex]y=\csc x{.}[/latex]

Complete the table and sketch a graph of [latex]y=\cot x{.}[/latex]

For Problems 59-64,

• Graph each function in the ZTrig window and write a simpler expression for the function.
• Show algebraically that your new expression is equivalent to the original one.

[latex]y=\dfrac{\csc x}{\cot x}[/latex]

[latex]y=\dfrac{\sec x}{\tan x}[/latex]

[latex]y=\dfrac{\sec x \cot x}{\csc x}[/latex]

[latex]y=\dfrac{\csc x \tan x}{\sec x}[/latex]

[latex]y=\tan x \csc x[/latex]

[latex]y=\sin x \sec x[/latex]

For Problems 65–70, find all solutions between [latex]0[/latex] and [latex]2\pi{.}[/latex]

[latex]3\csc \theta + 2=8[/latex]

[latex]-2\sec \theta + 7=3[/latex]

[latex]\sqrt{2}\sec \theta =-2[/latex]

[latex]8+\csc \theta =6[/latex]

[latex]2\cot \theta = -\sqrt{12}[/latex]

[latex]\sqrt{3}\cot \theta =1[/latex]

For Problems 71–76, use identities to find exact values or to write algebraic expressions.

If [latex]\tan \alpha = -2[/latex] and [latex]\dfrac{\pi}{2} \lt \alpha \lt \pi{,}[/latex] find [latex]\cos \alpha{.}[/latex]

If [latex]\cot \beta = \dfrac{5}{4}[/latex] and [latex]\pi \lt \beta \lt \dfrac{3\pi}{2}{,}[/latex] find [latex]\sin \beta{.}[/latex]

If [latex]\sec x = \dfrac{a}{2}[/latex] and [latex]0 \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\tan x{.}[/latex]

If [latex]\csc y = \dfrac{1}{b}[/latex] and [latex]\dfrac{\pi}{2} \lt y \lt \pi{,}[/latex] find [latex]\cot y{.}[/latex]

If [latex]\csc \phi = w[/latex] and [latex]\dfrac{3\pi}{2} \lt \alpha \lt 2\pi{,}[/latex] find [latex]\cos \phi{.}[/latex]

If [latex]\sec \theta = \dfrac{3}{z}[/latex] and [latex]\pi \lt \alpha \lt \dfrac{\pi}{2}{,}[/latex] find [latex]\sin \theta{.}[/latex]

For Problems 77–80, find exact values for [latex]\sec s,~ \csc s,[/latex] and [latex]\cot s{.}[/latex]

For Problems 81–88, write the expression in terms of sine and cosine, and simplify.

[latex]\sec \theta \tan \theta[/latex]

[latex]\csc \phi \cot \phi[/latex]

[latex]\dfrac{\csc t}{cot t}[/latex]

[latex]\dfrac{\tan v}{\sec v}[/latex]

[latex]\sec \beta - \tan \beta[/latex]

[latex]\cot \alpha + \csc \alpha[/latex]

[latex]\sin x \tan x - \sec x[/latex]

[latex]\csc y - \cos y \cot y[/latex]

Prove the Pythagorean identity [latex]1 + \tan^2 \theta = \sec^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\cos^2 \theta{.}[/latex])

Prove the Pythagorean identity [latex]1 + \cot^2 \theta = \csc^2 \theta{.}[/latex] (Hint: Start with the identity [latex]\cos^2 \theta + \sin^2 \theta = 1[/latex] and divide both sides of the equation by [latex]\sin^2 \theta{.}[/latex])

Suppose that [latex]\cot \theta = 5[/latex] and [latex]\theta[/latex] lies in the third quadrant.

• Use the Pythagorean identity to find the value of [latex]\csc \theta{.}[/latex]
• Use identities to find the values of the other four trig functions of [latex]\theta{.}[/latex]

Suppose that [latex]\tan \theta = -2[/latex] and [latex]\theta[/latex] lies in the second quadrant.

• Use the Pythagorean identity to find the value of [latex]\sec \theta{.}[/latex]

Write each of the other five trig functions in terms of [latex]\sin t[/latex] only.

Write each of the other five trig functions in terms of [latex]\cos t[/latex] only.

Show that if the angles of a triangle are [latex]A,~B,[/latex] and [latex]C[/latex] and the opposite sides are respectively [latex]a,~b,[/latex] and [latex]c,[/latex] then

[latex]a \csc A = b \csc B = c \csc C[/latex]

Trigonometry Copyright © 2024 by Bimal Kunwor; Donna Densmore; Jared Eusea; and Yi Zhen. All Rights Reserved.

Share This Book

Reciprocal Functions (Algebra 2 - Unit 8)

Also included in

Description

The Reciprocal Function Family Algebra 2 Lesson:

Your Pre-AP Algebra 2 Honors students will graph and anayze translations of reciprocal functions in this Unit 8 lesson. #distancelearningtpt

What is included in this resource?

⭐ Guided Student Notes

⭐ Google Slides®

⭐ Fully-editable SMART Board® Slides

⭐ Homework/Practice assignment

⭐ Lesson Warm-Up (or exit ticket)

⭐ Daily Content Quiz (homework check)

⭐ Video Lesson Link for Distance Learning - Flipped Classroom models

⭐ Full solution set

Students will be able to:

★ Graph reciprocal functions

★ Analyze and graph translations of reciprocal functions

Click HERE to SAVE 20% by buying all RATIONAL FUNCTIONS products, including cooperative activities, in UNIT 8 MEGA BUNDLE.

***(3/2/2022) Updated

The unit includes the following topics:

1) Inverse Variation

2) The Reciprocal Function Family

3) Rational Functions and Their Graphs

4) Rational Expressions

5) Adding and Subtracting Rational Expressions

6) Solving Rational Equations

7) Work and Motion Problem

You may also be interested in:

FOLDABLES ONLY

Guided Notes Only

SMART Board ONLY

Rational Functions Essentials

Activities & Assessments Bundle

Rational Functions Analyze Graph Activity

Tournament Review

Rational Functions Stations Activity

Here’s a FREE Product for this unit:

Rational Functions Four Square Activity FREEBIE

****************************************************************

I LOVE FEEDBACK

Remember to leave feedback and you will earn points toward FREE TPT purchases. I love that feedback!

Also, follow me to be notified about my new products, sales, updates, and FREEBIES!

If you have any questions, comments or special requests

please contact me by email at: [email protected]

Thanks for shopping in my store!

Questions & Answers

Flamingo math by jean adams.

• We're hiring
• Help & FAQ
• Privacy policy
• Student privacy
• Terms of service
• Tell us what you think

• Skip to main content

PAYMENT PLANS ARE NOW AVAILABLE • VISIT THE SHOP TO LEARN MORE!

All Things Algebra®

Algebra 2 Unit 8: Rational Functions

This unit includes 68 pages of guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the topics listed in the description below.

• Description
• Additional Information
• What Educators Are Saying

This unit contains the following topics:

• Simplifying Rational Expressions • Multiplying Rational Expressions • Dividing Rational Expressions • Adding and Subtracting Rational Expressions (Like Bases) • Adding and Subtracting Rational Expressions (Unlike Bases) • Simplifying Complex Fractions • Applications • Graphing Reciprocal Functions • Graphing Rational Functions • Identifying Key Characteristics: x-intercepts, vertical and horizontal asymptotes, holes, domain, range • Direct, Joint, Inverse, and Combined Variation

This unit does not contain activities.

This is the guided notes, homework assignments, quizzes, study guide, and unit test only.  For suggested activities to go with this unit, check out the ATA Activity Alignment Guides .

This resource is included in the following bundle(s):

Algebra 2 Curriculum

License Terms:

This purchase includes a single non-transferable license, meaning it is for one teacher only for personal use in their classroom and can not be passed from one teacher to another.  No part of this resource is to be shared with colleagues or used by an entire grade level, school, or district without purchasing the proper number of licenses.  A t ransferable license is not available for this resource.

Copyright Terms:

No part of this resource may be uploaded to the internet in any form, including classroom/personal websites or network drives, unless the site is password protected and can only be accessed by students.

What standards is this curriculum aligned to?

What format are the files in, will i have access to materials if they are updated, are answer keys included, are videos included.

This is one of the best units on rational functions with higher level thinking examples. The examples require more thought and effort from the students than many I have found. I love the entire unit!

The notes pages and practice problems are fantastic! Great examples. I love that the answer key is provided to assist both the teacher and students.

-JENNIFER S.

Excellent unit. I love the way notes are organized, the volume of practice problems provided, the quality quizzes & tests, and the video note links provided. Thanks for such a quality product!

-NICHOLE B.

You may also like these products

Order of Operations Spin to Win Game

Factoring Polynomials Spin to Win Game

Pumpkin Smash Game PowerPoint Template

Fraction & Decimal Operations Pumpkin Smash Bingo

Centers of Triangles Pumpkin Smash Bingo

Multi-Step, Compound, and Absolute Value Inequalities Pumpkin Smash Bingo

Multi-Step Inequalities Pumpkin Smash Bingo

Koosh Ball Game PowerPoint Template

One-Step Equations Koosh Ball Game

Exponent Rules Koosh Ball Game

Multi-Step Equations Koosh Ball Game

Circles (Arcs, Angles, Chords, Tangents, Secants) Koosh Ball Game

Fly-Swatter Game PowerPoint Template

Linear Equations Fly-Swatter Review Activity

Volume & Surface Area Fly-Swatter Bingo Game

Parallel Lines Cut by a Transversal Fly-Swatter Bingo Game

Proportions Fly-Swatter Bingo Game

Probability (Simple and Compound) Fly-Swatter Bingo Game