Lesson 24 Solving Multi
Essential QuestionHow can you solve a multi-step inequality?
EXPLORATION 1 Solving a Multi-Step Inequality Work with a partner.
Use what you already know about solving equations and inequalities to solve each multi-step inequality. Justify each step. Match each inequality with its graph. Use a graphing calculator to check your answer.a. 2x + 3 x + 5b. -2x + 3 > x + 9c. 27 5x + 4xd. -8x + 2x 16 < -5x + 7xe. 3 5x > -3x 6f. -5x 6x 8 8x xAnswer:a. 2x + 3 x + 5b. -2x + 3 > x + 9c. 27 5x + 4xd. -8x + 2x 16 < -5x + 7xe. 3 5x > -3x 6f. -5x 6x 8 8x xThe given graphing calculators are:Now,From the graphing calculators, we can observe that the graph is divided into 4 parts.The first part indicates +xThe second part indicates -xNow,
We can conclude that the solution to the given inequality is x -4The graph F) matches the solution of the given inequality
Question 2.How can you solve a multi-step inequality?Answer:The general procedure for solving multi-step inequality is as follows:a) Clear parenthesis i.e., Brackets on both sides of the inequality and collect like termsb) Addor subtract terms so the variable is on one side and the constant is on another side of the inequality sign
We can conclude that the given inequality has no solution
Question 31.MODELING WITH MATHEMATICSWrite and solve an inequality that represents how many $20 bills you can withdraw from the account without going below the minimum balance.Answer:
Characteristics Of Quadratic Functions 22 Exercises
Vocabulary and Core Concept and Check
Question 1.WRITING Explain how to determine whether a quadratic function will have a minimum value or a maximum value.Answer:
Question 2.WHICH ONE DOESNT BELONG? The graph of which function does not belong with the other three? Explain.Answer:
y = \x2 3x + 6Answer:
y = \x2 4x 1Answer:
Question 31.WRITING Two quadratic functions have graphs with vertices and . Explain why you can not use the axes of symmetry to distinguish between the two functions.Answer:
Question 32.WRITING A quadratic function is increasing to the left of x = 2 and decreasing to the right of x = 2. Will the vertex be the highest or lowest point on the graph of the parabola? Explain.Answer:
ERROR ANALYSIS In Exercises 33 and 34, describe and correct the error in analyzing the graph of y = 4×2 + 24x 7.
MODELING WITH MATHEMATICS In Exercises 35 and 36, x is the horizontal distance and y is the vertical distance . Find and interpret the coordinates of the vertex.
Question 35.The path of a basketball thrown at an angle of 45° can be modeled by y = -0.02×2 + x + 6.Answer:
Question 36.The path of a shot put released at an angle of 35° can be modeled by y = -0.01×2 + 0.7x + 6.Answer:
Question 37.ANALYZING EQUATIONS The graph of which function has the same axis of symmetry as the graph of y = x2 + 2x + 2?A. y = 2×2 + 2x + 2B. y = -3×2 6x + 2C. y = x2 2x + 2D. y = -5×2 + 10x + 23Answer:
y = \x2 3x + 2Answer:
Lesson 24 Modeling With Quadratic Functions
Essential QuestionHow can you use a quadratic function to model a real-life situation?
EXPLORATION 1Modeling with a Quadratic Function
Work with a partner. The graph shows a quadratic function of the formP = at2 + bt + cwhich approximates the yearly profits for a company, where P is the profit in year t.a. Is the value of a positive, negative, or zero? Explain.b. Write an expression in terms of a and b that represents the year t when the company made the least profit.c. The company made the same yearly profits in 2004 and 2012. Estimate the year in which the company made the least profit.d. Assume that the model is still valid today. Are the yearly profits currently increasing, decreasing, or constant? Explain.
EXPLORATION 2Modeling with a Graphing CalculatorWork with a partner. The table shows the heights h of a wrench t seconds after it has been dropped from a building under construction.a. Use a graphing calculator to create a scatter plot of the data, as shown at the right. Explain why the data appear to fit a quadratic model.b. Use the quadratic regression feature to find a quadratic model for the data.c. Graph the quadratic function on the same screen as the scatter plot to verify that it fits the data.d. When does the wrench hit the ground? Explain.
Communicate Your Answer
How can you use a quadratic function to model a real-life situation?
Question 1.WHAT IF? The vertex of the parabola is . What is the height of the net?
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Free Algebra 1 Diagnostic Tests
Algebra I focuses on solving quadratic functions using the quadratic formula and FOIL, as well as graphing parabolas and manipulating their appearance through changes made to the source equation. Other concepts that may be introduced at various points within Algebra I classes are statistics and probability, percent, and percent change. While not directly related to the overarching ideas of equations, functions, and graphs, these concepts may be taught in a way that mirrors the back-and-forth logic used to teach students about functions and their graphs. For instance, a focus when learning percents is how to convert a percent to a decimal and vice versa, and when expressing the probability of an event occurring, students also necessarily figure out the probability of the event not occurring. The mathematical concepts that students master in Algebra I form the core of their mathematical understanding in many later classes in math and the sciences.
Solving Linear Inequalities Study Skills: Analyzing Your Errors
2.12.4 What Did You Learn?
Representing Linear Inequalities, p. 57
Multiplication and Division Properties of Inequality , p. 68Multiplication and Division Properties of Inequality , p. 69
Section 2.4Solving Multi-Step Inequalities, p. 74Special Solutions of Linear Inequalities, p. 75
Question 1.Explain the meaning of the inequality symbol in your answer to Exercise 47 on page 59. How did you know which symbol to use?Answer:In Exercise 47 on page 59,The inequality symbol we used is: The meaning of is Less than or equal to In Exercise 47,It is given that the Xianren bridge arch is the longest natural arch with a length of 400 feet i.e., there is no arch longer than the Xianren bridge arch and the remaining natural arches are shorter than the Xianren archHence,The lengths of all the arches including the Xianren arch will be represented by the inequality symbol
Question 2.In Exercise 30 on page 66, why is it important to check the reasonableness of your answer in part before answering part ?Answer:In part , it is given that you have to beat your competitor with your score.So,Your score must be greater than your competitorThen only you can solve part .
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Quadratic Functions 21 22 Quiz
2.1 2.2 Quiz
Describe the transformation of f = x2 represented by g.
Write a rule for g and identify the vertex.
Question 4.Let g be a translation 2 units up followed by a reflection in the x-axis and a vertical stretch by a factor of 6 of the graph of f = x2.
Question 5.Let g be a translation 1 unit left and 6 units down, followed by a vertical shrink by a factor of \ of the graph of f = 32.
Question 6.Let g be a horizontal shrink by a factor of \, followed by a translation 1 unit up and 3 units right of the graph of f = 2 11.
Graph the function. Label the vertex and axis of symmetry.
g = \
Question 12.f = 0.4x
Question 13.A grasshopper can jump incredible distances, up to 20 times its length. The height of the jump above the ground of a 1-inch-long grasshopper is given by h = \x2 + x, where x is the horizontal distance of the jump. When the grasshopper jumps off a rock, it lands on the ground 2 inches farther. Write a function that models the new path of the jump.
Question 14.A passenger on a stranded lifeboat shoots a distress flare into the air. The height of the flare above the water is given by f = -16t, where t is time since the flare was shot. The passenger shoots a second flare, whose path is modeled in the graph. Which flare travels higher? Which remains in the air longer? Justify your answer.
Lesson 21 Transformations Of Quadratic Functions
How do the constants a, h, and k affect the graph of the quadratic function g = a2 + k?The parent function of the quadratic family is f = x2. A transformation of the graph of the parent function is represented by the function g = a2 + k, where a 0.
EXPLORATION 1Identifying Graphs of Quadratic FunctionsWork with a partner. Match each quadratic function with its graph. Explain your reasoning. Then use a graphing calculator to verify that your answer is correct.a. g = -2b. g = 2 + 2c. g = -2 2d. g = 0.52 + 2e. g = 22f. g = -2 + 2
Communicate Your Answer
Question 2.How do the constants a, h, and k affect the graph of the quadratic function g =a2 + k?
Question 3.Write the equation of the quadratic function whose graph is shown at the right. Explain your reasoning. Then use a graphing calculator to verify that your equation is correct.
Describe the transformation of f = x2 represented by g. Then graph each function.
g = -2 + 2
Question 7.Let the graph of g be a vertical shrink by a factor of \ followed by a translation 2 units up of the graph of f = x2. Write a rule for g and identify the vertex.
Question 8.Let the graph of g be a translation 4 units left followed by a horizontal shrink by a factor of \ of the graph of f = x2 + x. Write a rule for g.
Question 9.WHAT IF? In Example 5, the water hits the ground 10 feet closer to the fire truck after lowering the ladder. Write a function that models the new path of the water.
Algebra 2 Chapter 3 Answers
The Chapter 6 Resource Mastersincludes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. All of the materials found in this booklet are included for viewing, printing, and editing Mid-Chapter Test This 1-page test provides an option to assess the first half of All of the materials found in this booklet are included for viewing, printing, and Then find the roots and graph the related function. Real Estate A developer wants to Part I 23C x 3 undefined? The topics covered in this chapter are factorials, permutations, fundamentals of counting, probability of compound and Chapter 6 51 Glencoe Algebra 1.
Quadratic Functions Chapter Test
Question 1.A parabola has an axis of symmetry y= 3 and passes through the point . Find another point that lies on the graph of the parabola. Explain your reasoning.
Question 2.Let the graph of g be a translation 2 units left and 1 unit down, followed by a reflection in the y-axis of the graph of f = 2 4. Write a rule for g.
Question 3.Identify the focus, directrix, and axis of symmetry of x = 2y2. Graph the equation.
Question 4.Explain why a quadratic function models the data. Then use a linear system to find the model.
Write an equation of the parabola. Justify your answer.
Question 8.A surfboard shop sells 40 surfboards per month when it charges $500 per surfboard. Each time the shop decreases the price by $10, it sells 1 additional surfboard per month. How much should the shop charge per surfboard to maximize the amount of money earned? What is the maximum amount the shop can earn per month? Explain.
Question 9.Graph f = 8×2 4x+ 3. Label the vertex and axis of symmetry. Describe where the function is increasing and decreasing.
Question 10.Sunfire is a machine with a parabolic cross section used to collect solar energy. The Suns rays are reflected from the mirrors toward two boilers located at the focus of the parabola. The boilers produce steam that powers an alternator to produce electricity.a. Write an equation that represents the cross section of the dish shown with its vertex at .b. What is the depth of Sunfire? Justify your answer.
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Lesson 13 Modeling With Linear Functions
Essential QuestionHow can you use a linear function to model and analyze a real-life situation?
EXPLORATION 1Modeling with a Linear FunctionWork with a partner. A company purchases a copier for $12,000. The spreadsheet shows how the copier depreciates over an 8-year period.a. Write a linear function to represent the value V of the copier as a function of the number t of years.b. Sketch a graph of the function. Explain why this type of depreciation is called straight line depreciation.c. Interpret the slope of the graph in the context of the problem.
EXPLORATION 2Modeling with Linear FunctionsWork with a partner. Match each description of the situation with its corresponding graph. Explain your reasoning.a. A person gives $20 per week to a friend to repay a $200 loan.b. An employee receives $12.50 per hour plus $2 for each unit produced per hour.c. A sales representative receives $30 per day for food plus $0.565 for each mile driven.d. A computer that was purchased for $750 depreciates $100 per year.
Communicate Your Answer
How can you use a linear function to model and analyze a real-life situation?
Question 4.Use the Internet or some other reference to find a real-life example of straight line depreciation.a. Use a spreadsheet to show the depreciation.b. Write a function that models the depreciation.c. Sketch a graph of the function.
Algebra 2 Chapter 1 Test Answer Key
Algebra 2 chapter 1 test answer key Alg 2 Chapter 1 HW – Answers OnlyFile Size: 1480 kbFile Type: pdfDownload File Think Book Table of Contents Chapter 1 & Algebra Review Size: 265 kbFile Type: pdfDownload File Worked HW Solutions 1-4 to 1-18File Size: 244 kb File type: pdfCrito reload Math Note Entry with Interaction think
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Solving Compound Inequalities 25 Exercises
Vocabulary and Core Concept Check
Question 1.WRITINGCompare the graph of -6 x -4 with the graph of x -6 or x -4.Answer:
WHICH ONE do DOESNT BELONG?Which compound inequality does not belong with the other three? Explain your reasoning.Answer:a. a > 4 or a < -3b. a< -2 or a > 8c. a > 7 or a < -5d. a < 6 or a > -9Now,Represent all the inequalities in the graphSo,
From the above graph,We can observe thatThe first marked line started from 4 including 4 and continued till the end of the left end of the graphThe second marked line started from 6 excluding 6 and continued till the end of the right end of the graphHence,The representation of the given graph in the form of inequality is:x 4 or x > 6
In Exercises 710, write the sentence as an inequality. Graph the inequality.
Question 7.A number p is less than 6 and greater than 2.Answer:
A number n is less than or equal to -7 or greater than 12.Answer:The given worded form is:A number n is less than or equal to -7 or greater than 12Hence,The representation of the given worded form in the form of inequality is:n -7 or n > 12The representation of the inequalities in the graph is:
Question 9.A number m is more than -7\ or at most -10.Answer:
Question 11.MODELING WITH MATHEMATICSSlitsnails are large mollusks that live in deep waters. They have been found in the range of elevations shown. Write and graph a compound inequality that represents this range.Answer:
In Exercises 1320, solve the inequality. Graph the solution.
Lesson 23 Focus Of A Parabola
Essential QuestionWhat is the focus of a parabola?EXPLORATION 1Analyzing Satellite DishesWork with a partner. Vertical rays enter a satellite dish whose cross section is a parabola. When the rays hit the parabola, they reflect at the same angle at which they entered. a. Draw the reflected rays so that they intersect the y-axis.b. What do the reflected rays have in common?c. The optimal location for the receiver of the satellite dish is at a point called the focus of the parabola. Determine the location of the focus. Explain why this makes sense in this situation.
EXPLORATION 2Analyzing SpotlightsWork with a partner. Beams of light are coming from the bulb in a spotlight, located at the focus of the parabola. When the beams hit the parabola, they reflect at the same angle at which they hit. Draw the reflected beams. What do they have in common? Would you consider this to be the optimal result? Explain.
Communicate Your Answer
What is the focus of a parabola?
Question 4.Describe some of the properties of the focus of a parabola.
Question 1.Use the Distance Formula to write an equation of the parabola with focus F and directrix y = 3.
Identify the focus, directrix, and axis of symmetry of the parabola. Then graph the equation.
Write an equation of a parabola with vertex and focus .
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Transformations Of Linear And Absolute Value Functions 12 Exercises
Vocabulary and Core Concept Check
Question 1.The function g = | 5x |- 4 is a horizontal ___________ of the function f = | x | 4.Answer:
Question 2.WHICH ONE DOESNT BELONG? Which transformation does not belong with the other three? Explain your reasoning.Answer:
Monitoring Progress and Modeling with Mathematics
In Exercises 38, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.
Question 3.f = x 5 translation 4 units to the leftAnswer:
f = x + 2 translation 2 units to the rightAnswer:
f = | 4x + 3 | + 2 translation 2 units downAnswer:
f = 2x 9 translation 6 units upAnswer:
Question 9.WRITING Describe two different translations of the graph of f that result in the graph of g.Answer:
Question 10.PROBLEM SOLVING You open a café. The function f = 4000x represents your expected net income after being open x weeks. Before you open, you incur an extra expense of $12,000. What transformation of f is necessary to model this situation? How many weeks will it take to pay off the extra expense?Answer:
In Exercises 1116, write a function g whose graph represents the indicated transformation of the graph of f. Use a graphing calculator to check your answer.
Question 11.f = -5x+ 2 reflection in the x-axisAnswer:
f = \x 3 reflection in the x-axisAnswer:
f = | 6x | 2 reflection in the y-axisAnswer:
f = | 2x 1 | + 3 reflection in the y-axisAnswer:
f = -2 | x 4 | + 2Answer: