Pre Calculus 11 Absolute Value And Reciprocal Functions Test

BC’s New Curriculum (Big Ideas, Curricular and Core Competencies) for Pre-calculus 11 can be accessed at:. Reduce the rational function to lowest terms, if possible. This is necessary because the definition of a function requires that for every x there exists exactly one y. Section (Assignment) Absolute Value and Reciprocal Functions TEST. Click on any files in that folder to open them. Test and improve your knowledge of MTEL Math: Piecewise, Absolute Value & Step Functions. 2 Solving Absolute Value Equations Checkpoint 8. 2: Absolute Value Functions 7. 6 CHAPTER 1 PRE-CALCULUS REVIEW Try it. Typically, students have previously attained knowledge of pre-calculus topics (some combination of trigonometry, elementary functions, analytic geometry, and math analysis). The logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x. 2 Absolute Value Equations 638 4 - 11, 13 8. Textbook: Teacher generated notes. Sarah Eichorn and Dr. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Absolute Value Function. Interactive solvers for algebra word problems. The absolute value function is commonly used to measure distances between points. Functions: Simplifying Difference Quotients* 6. 2 Functions and Their Properties PreCalculus 1 - 2 For many functions, the domain is all real numbers, or. Graph of a Function In case both the inputs and outputs of a function are numbers, we can draw a. pdf: File Size: 672 kb: File Type: pdf. Consider 9x1/9 = 1 9 is the. 3 Absolute Value Equations Pg. The easiest way to think of them is if you drew more than one function on a graph, and you just erased parts of the functions where they aren’t supposed to be (along the \(x\)’s); they are defined differently for different intervals of \(x\). NO TEXTBOOK REQUIRED. 4) Graphing Inequalities & Systems of Equations (Ch. For instance, 3 is three away from 0. We are using Theory and Problems for Pre-Calculus Mathematics 11 workbook. The key concepts covered in Pre-Calculus 11 are as follows: Algebra and Number - absolute value, operations on radical numbers and expressions, solving and applying radical equations, using rational expressions, solving and applying rational equations; Geometry - angles in the standard position, trigonometric ratios, using the sine and. Absolute Value and Reciprocal Functions Key Terms absolute value absolute value function piecewise function invariant point absolute value equation reciprocal function asymptote. B: Protect the second input of the rem function from going to zero. [2 marks] 15) Solve x2 2x 6 [3 marks] Domain for Case 1: _____. Click Groups. A = 1 2 bh = 1 2 | |30−0 ( )5 = 75 The triangle has an area of 75 square units. Functions dealing with floating-point representation. 410 #2, 3, 6abc, 7abc, 8, 11, 12 Reciprocals: p. Here are the steps required to find the inverse function : Step 1: Determine if the function has an inverse. Graphing Absolute Value Function Translations Graphing Absolute Value Functions: Reflection and Translation Graphing Absolute Value Functions: Transformation Graphing Absolute Value Functions: Transformation y=-1/2|x-3|+4 Ex 1: Graph a Transformation of an Absolute Value Function Using a Table Ex 2: Graph a Transformation of an Absolute Value. Suderman's Math Website: Menu Pre-Calculus 11 Foundations of Math 11 Pre-Calculus 12 Ch. 2: Absolute Value Functions 7. Sketch a graph of the function. Absolute Value and Reciprocal Functions 8. 7: Absolute Value & Reciprocal Functions. 5 quadratic functions & equations. Chapter 3 - Quadratic Functions. deferred until Section 2. 2: Graphing Absolute Value Functions • Section 7. 3 Dividing Polynomial Functions 2. 5 Graphing Reciprocals of Quadratic Functions Study Guide Review Practice Test Cumulative Review. Systems of Equations and Inequalities Chapter 7. 6 CHAPTER 1 PRE-CALCULUS REVIEW Try it. 2 Absolute Value & Functions | WS Key. Workbook fee is nonrefundable. o Determine the domain and range of functions in general, including root and rational functions. We are going to round things out and look at all the parent functions and sort of formalize our analysis with more complicated examples. Tutorials for Precalculus Demana text. 3 trigonometry (part a - no calculator) test - ch. perpendicular to the line 7x-6y+11=0 ? On what interval(s) is the function increasing? Find the maximum value of the function. Chapter 5 - Functions and Equations. 3 Graphing Reciprocals of Linear Functions 8. The real part of a complex number Z is denoted as Re (Z). Use this worksheet/quiz to find out how much you know about reciprocal functions. I'm just trying to determine what the equations for the inverse of the absolute function is. Along the way, students see how the algebra connects to their future calculus courses, with tools like Foreshadowing Calculus. 1 3 prwl precalculus how to test determine if a function is odd even or neither finding the absolute value of complex numbers 1. 4 credits Prerequisite: Math Placement Test score of 7 or higher or MATH103 (C or better) Discrete Mathematics. Pre-Calculus. I answer questions and make sure that students see know the value of the left endpoint for Questions 2 and 3 is undefined. Topic Chapters Sequence & Series 1 Trigonometry 2 Quadratic Functions 3 Quadratic Equations 4 Radical Expressions & Equations 5 Rational Expressions & Equations 6 Absolute Value & Reciprocal Fns 7 Systems of Equations 8. functions. So, absolute value of -8 is 8. This will take you to a list of folders. [2 marks] 15) Solve x2 2x 6 [3 marks] Domain for Case 1: _____. MHF4U Advanced Functions. Pre-Calculus 11 and 12 is designed for students who intend to pursue a career in mathematics, sciences, or engineering. vertical stretch n , (of a graph) the result of replacing each point of the graph with the point obtained by scaling the \(y\)-coordinate by a fixed factor (when that factor is. The absolute value is used in the real world to define the DIFFERENCE or change from one point to another. Identify the effect on the graph of replacing f (x) by f (x) + k, k f(x), f (kx), and f (x + k) for specific values of k given the graphs. You will put. 5 Graphing Reciprocals of Quadratic Functions Review & Quiz Unit Test. Mathematics 11 curriculum. SCO RF5: Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions). Start with f(1). Greatest Integer Function yxx int (Step Function) x y 12. Click Groups. 8 Combining and Composite Functions. Absolute Value Formulas for Pre-Calculus. The best videos and questions to learn about Transformations of the Reciprocal Function. Ahearn, Kevin (Social Science) Alexander, Jennifer (Reading Specialist) Amor, Marina (World Languages and Cultures) Astorga, Angela (English) Augustowski, Jason (English). 1 Absolute value of real numbers, equations and functions (linear and quadratic). Title your page with the textbook section and page numbers. It takes time to study for a comprehensive mathematics exam; time to review your notes, time to practice a wide variety of problems, and time before and after school. mid-term exam (part a – no calculator) test – ch. Kenneth Senstad. 2) Sketch and describe the properties of absolute functions. However, if a function is restricted to a certain domain so that it passes the horizontal line test, then in that restricted domain, it can have an inverse. Set b and c to zero and select a positive value for a to obtain a quadratic function. 4 Notes February 11th Chapter Quiz February 13th Fermat Day February 18th Chapter Test. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Absolute value describes the distance of a number on the number line from 0 without considering which direction from zero the number lies. y x 4 2 2 4 4 x 0. They both would fail the horizontal line test. Topics include algebra of the real numbers, algebraic, exponential, and logarithmic functions and their graphs, systems of equations, inequalities, and absolute value. Students will study and write proofs and learn about the Law of Sines and Cosines and their applications. For courses in Precalculus. CC+ College Algebra. 2 Part 2 Notes February 5th 7. Pearson: Pre-calculus 11: my WORKTEXT ISBN-13 978--321-62449-9 The workbook can be purchased ($30) at either one of our locations. Pre-Calc 11 Preview provides a course outline, including a general timeline. Remember the same y value can be assigned to more than one x value in a function. 52 Chapter 8: Absolute Value and Reciprocal Functions—Practice Test—Solutions DO NOT COPY. Pre Calculus Properties of Quadratic Functions. 4 Solve absolute value equations. Verify inverse functions. Quadratic Inequality in One Variable using Test Intervals. 2 Solving Absolute Value Equations Checkpoint & Quiz 8. Chapter 5 - Functions and Equations. deferred until Section 2. The first test can be written from home, but the remaining tests must be. 2 Absolute Value & Functions | WS Key. A sampling of data for the identity function is presented in tabular form below:. ) Applications of Trigonometry (Periodic Motion) Applications of Trigonometry (Triangles) Basic Trig Functions Exponential Functions Exponential Growth and Decay Function Composition Functions Graphs of Trigonometric Functions Introduction to Periodic Motion. 3 (Part 2) Solving Compound Inequalities - Special Cases - Module 2. 3 Reciprocals of Linear Functions 657 3, 5 - 7, 9, 11, 16 Lab 8. Created by our FREE tutors. Pre Calculus Text Book. Great lesson for Math Analysis or pre-calculus students and teachers. Practice Tests by Teodoru Gugoiu. It has a corner point at which the graph changes direction. Pre-Calculus 11- Course Outline: File Size: Quadratic Functions. Pre-Calculus 11. Focus On Chapter 7 7. com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus. Applications with Absolute Value Inequalities - Mod 2. As you progress, you'll also conquer topics such as absolute value, nonlinear inequalities, inverses, trigonometric functions, and conic sections. 1 Sequences & Series Notes from Thursday, May 23rd: Reciprocal Function Notes. Thursday – Chapter 2B Whiteboard Review. 4_Linear_Reciprocal_Functions. Graphing Trig Functions Practice Worksheet With Answers Students will practice graphing sine and cosine curves : a) identify period and amplitude based on equation or on the graph b) write equation from graph c) write. Students taking grade 11 and 12 math may decide to enroll in one of three math branches: Foundations of Math, Pre-Calculus, or Apprenticeship and Workplace. a) Determine its reciprocal function. Pre-algebra, Algebra I, Algebra II, Geometry, Physics. The real short way that absolute values work is that if you have f(x) graphed the absolute value of f(x) has it so that when the y value is about to become less than zero the value then bounces so that y is always positive. Substituting 0 for x, you find that cos x approaches 1 and sin x − 3 approaches −3; hence,. 1: Defining and Evaluating Absolute Values • Section 7. Course Outline Unit 3: Absolute Value & Reciprocal Functions. quantities involving functions and their inverses RFR. The Absolute Value Function. Piecewise functions (or piece-wise functions) are just what they are named: pieces of different functions (sub-functions) all on one graph. Multiply (if necessary). Computers use binary arithmetic with finite precision. Algebra 2 Team Tutorials – If I am not available, try one of the other great Algebra 2 teachers. Here’s how you can test the circles and semi-circle functions. Click HERE to access another set of videos that you might prefer. The timeline is not set. Chapter 3 - Quadratic Functions. Pearson: Pre-calculus 11: my WORKTEXT ISBN-13 978--321-62449-9 The workbook can be purchased ($30) at either one of our locations. f (x) = {x if x > 0 0 if x = 0 − x if x < 0. Topics include asymptotes and graphing, intercepts, and domain / range. Practice Test #1-15 #10 replace 19d. The period, Module 8: Absolute Value and Reciprocal Functions (19 marks) 1. 04 at x = -1. Pre Calculus Text Book. 2: Absolute Value Functions 7. 4 Math Lab: Using Technology to Graph Reciprocals of Quadratic Functions 8. 2 Systems of Linear Equations in 3 Variables. Pre-Calculus 11 Ms. Exponential Function yex 8. Absolute Value Functions Inverse and Reciprocal Trigonometric Functions (Old Study Design Questions). 7 absolute value functions & reciprocal functions. Subtract one number from the other and give the result the sign of the number that has the greater absolute value. , absolute value, radical, reciprocal, conics, exponential, logarithmic, trigonometric) — vertical and horizontal translations, stretches, and reflections — inverses: graphs and equations. Pre-Calculus 11 Assignments Date Chapter Lesson Assignment Math 10 Review, Course Outline, Texts test Unit 6 test 7. Kenneth Senstad. Absolute Value Functions Algebra Review (fractions, factoring, simplification, etc. 1 Absolute Value Functions 8. Don't like my videos (sniff! sniff!). Do I just take the positive version of it then solve and then the negative version and solve?. Pre-Calc 11 Preview provides a course outline, including a general timeline. PreCalculus Honors: Functions and Their Graphs Semester 2 , Unit 5: Activity 26 Resources: SpringBoard- In this unit, PreCalculus points and curves in polar form and relate these to their rectangular form. Title your page with the textbook section and page numbers. Where does the absolute value function y. Test: Mon, Feb 25 (block 5), Tues, Feb 26 (block 1) Day 1: Absolute Value Day 2: Absolute value functions Day 3-4: Absolute Value Equations Day 5-6: Reciprocal Functions Day 7: Quiz and Review package and p. Pre-Calculus 11 requires that students have completed Principles of Math 10 or Foundations of Math and Pre-Calculus 10. use the trigonometric functions to solve right triangles. Jasper Place is the third school (and third city Airdrie and Calgary being the other two) where I've taught. o Identify and graph linear, absolute value, quadratic, cubic, and square root functions and their transformations. Trigonometric Functions Algebra All Content Math. PRE-CALCULUS 11 - Exam Outline Absolute Value and Reciprocal Functions o Evaluating absolute value expressions o Graphing absolute value functions (linear and. This is necessary because the definition of a function requires that for every x there exists exactly one y. 2: Absolute Value Functions 7. Online precalculus video lessons to help students with the notation, theory, and problems to improve their math problem solving skills so they can find the solution to their Precalculus homework and worksheets. Use the graph of the reciprocal function to graph the linear function. Tests: At the end of each unit there is a test, which is to be written in the classroom under supervision. In this case the absolute value will be zero if \(z = 6\) and so the absolute value portion of this function will always be greater than or equal to zero. Precalculus, Lecture 11, Part 4, getting Information From a graph Precalculus, Lecture 11, Part 5, Graphs of Piecewise Defined Functions Precalculus, Lecture 11, Part 6, Vertical Line Test. , one such that as the value of x approaches any given value a, the value of f(x) approaches that of f(a) as a limit. W2015 Abs Value Test. 8 Demonstrate understanding of quadratic equations). Applications with Absolute Value Inequalities - Mod 2. Precalculus Notes: Unit 1 – Functions Page 11 of 24 Precalculus – Graphical, Numerical, Algebraic: Pearson Chapter 1 7. 3 Notes, Lesson 7. Welcome to Grade 11 Pre-Calculus Mathematics! This course is a continuation of the concepts you have studied in previous years, as well as an introduction to new topics. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Computers use binary arithmetic with finite precision. Determine the domain and range of an inverse function, and restrict the domain of a function to make it one-to-one. The book costs $25. The reciprocal functions, sine and cosine, are easier to graph because they don’t have as many complex parts (no asymptotes, basically). Chapter 7 Absolute Value and Reciprocal Functions outline for unit Skills required for this unit: Graphing y=mx+b (math 10), Graphing y=a(x-p)2+q vertex form (skill 9 and 10), Solving by Isolating x (skill 16), by Factoring (Skill 15) and by Quadratic Formula (Skill 17), Determining the vertex (Skill 11) 26) Notes Evaluate Absolute Value 27) Notes Graphing…. Chapter 7: Absolute Value and Reciprocal Functions Section 7. Pre-Calculus 11 and 12 is designed for students who intend to pursue a career in mathematics, sciences, or engineering. 2 Solving Absolute Value Equations Checkpoint & Quiz 8. Polynomial, Rational, and Power Functions 2. 8 Demonstrate understanding of quadratic equations). The real short way that absolute values work is that if you have f(x) graphed the absolute value of f(x) has it so that when the y value is about to become less than zero the value then bounces so that y is always positive. Absolute value—vertical shift up 5, horizontal shift right 3. Ppt 5 3 Solving Trigonometric Equations Powerpoint. On what intervals is the function decreasing? 25. and so does the reciprocal function, because graph the absolute value function. 4 Math Lab: Using Technology to Graph Reciprocals of Quadratic Functions 8. _7_-_extra_review_-_solutions. Systems of Equations and Inequalities Chapter 7. ulp and Math. Pre Calculus Text Book. - Demonstrate an understanding of the absolute valueof real numbers. On this page you will find: a complete list of all of our math worksheets relating to Precalculus. What happens to the value of y when x is close to a 5. Chapter 7 : Absolute Value & Reciprocal Functions Calculus 12 Course Review. Rational Expressions Absolute Value and Reciprocal Functions. Rachel Lehman is licensed under a Creative Commons Attribution-ShareAlike 4. Rational Functions: Increasing and Decreasing Revisited 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. It has a corner point at which the graph changes direction. 5 Absolute Value Transformations 2014. 2: Graphing Absolute Value Functions • Section 7. Students will study and write proofs and learn about the Law of Sines and Cosines and their applications. Reciprocal Function Characteristics. absolute value and step functions common core algebra 1 homework answersabsolute value and step functions common core algebra i homework answers cd4164fbe1 Common Core Algebra I;. - The domain of a function is the set of all values for which the function is defined. 1: Defining and Evaluating Absolute Values • Section 7. Absolute Value Formulas for Pre-Calculus. 4 Homework _ (bcmath) 7. f x x 2 5. Lesson 1-7 – Translating Absolute Value Functions Discovery Worksheet (with answers) Name Date Graphing Absolute Value Functions Graph the following piecewise function by hand: (1) 0 0 x if x fx x if x (2) On your graphing calculator graph the function f(x) x with this WINDOW and answer the following questions. Do I just take the positive version of it then solve and then the negative version and solve?. The absolute value function is commonly used to measure distances between points. Example Items. If we designate the absolute value of an algebraic expression such as $$\mid x+1 \mid$$ and x has a value such that x+1 is a negative number, then the absolute value of the expression will be negative of the expression. Pre-Calculus 11 Chapter 7 Absolute Value and Reciprocal Functions. 1, Lesson 7. mid-term exam (part a – no calculator) test – ch. 3 Absolute Value Equations 7. Absolute Value Helps When the Distance is Negative. Precalculus tutors Solve it with our pre-calculus. 6 Transformations of Functions. trigonometry (angles in standard position, non-right triangles), absolute value functions, reciprocal functions, quadratic functions and equations, systems of equations and inequalities (including quadratic), and arithmetic and geometric sequences. They both would fail the horizontal line test. Lesson 1-7 – Translating Absolute Value Functions Discovery Worksheet (with answers) Name Date Graphing Absolute Value Functions Graph the following piecewise function by hand: (1) 0 0 x if x fx x if x (2) On your graphing calculator graph the function f(x) x with this WINDOW and answer the following questions. 5 Complex Zeroes of Polynomial Functions and the Fundamental Theorem of Algebra 2. 2 - Arithmetic Combinations of Functions Functions are number generators. 1 Absolute Value 7. Rational Expressions Absolute Value and Reciprocal Functions. The best videos and questions to learn about Transformations of the Reciprocal Function. You will also learn how they are used to solve problems. 2 Notes on Graphing Reciprocal Functions Reciprocal Linear and Quadratic Functions Worksheets Jan 7 Quiz 7. Coördinate pairs of a function. Subtract one number from the other and give the result the sign of the number that has the greater absolute value. Practice Test #1-15 #10 replace 19d. You'll gradually master functions, graphs of functions, logarithms, exponents, and more. Absolute Value and Reciprocal. Enter code: 4mcmv You can also join the class by using this link:. The however advanced, can permit division by zero. Powered by Create your own unique website with customizable templates. 3we de ned a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. Definition. Pre-Calculus 11. Trigonometric Functions Algebra All Content Math. Example 1 Graphing a Piecewise Function. 1_absolute_value_functions__complete_. 4) Graphing Inequalities & Systems of Equations (Ch. Consider 9x1/9 = 1 9 is the. Great lesson for Math Analysis or pre-calculus students and teachers. 2 Assignment Page 375 Questions 1-11 7. of items for the ACP. ryerson pre. Precalculus: Mathematics for Calculus, 7th Edition answers to Chapter 2 - Section 2. Pre-Calculus Mathematics 12 - 1. 3 Section 7. 4: Graphing Reciprocal Functions • Section 7. 8 R E A L L I F E Graphs of Absolute Value Functions. Students are encouraged to take an algebraic approach with the graphical approach as a secondary source. 9 Inverse Functions. For the following exercises, use the graph of the piecewise function shown below. f x = x 2 if x ≤ 1 3 if 1 < x ≤ 2 x if x > 2. On what intervals is the function decreasing? 25. Pre-Calculus 11 Unit 8: Absolute and Reciprocal Lesson 8. 2 Notes on Graphing Reciprocal Functions Reciprocal Linear and Quadratic Functions Worksheets Jan 7 Quiz 7. These courses are to be taken consecutively, not concurrently. Given the equations of two lines, determine whether their graphs are parallel or perpendicular. Beyond simple math and grouping (like "(x+2)(x-4)"), there are some functions you can use as well. Take one of our many Precalculus practice tests for a run-through of commonly asked questions. Jasper Place is the third school (and third city Airdrie and Calgary being the other two) where I've taught. Graphing Absolute Value Function Translations Graphing Absolute Value Functions: Reflection and Translation Graphing Absolute Value Functions: Transformation Graphing Absolute Value Functions: Transformation y=-1/2|x-3|+4 Ex 1: Graph a Transformation of an Absolute Value Function Using a Table Ex 2: Graph a Transformation of an Absolute Value. Year 12 Specialist Maths. Time4Learning offers an online, interactive, high school math curriculum that is organized into five courses that correlate to state standards: Algebra 1, Geometry, Algebra 2, Trigonometry, and Pre-Calculus. 3 Graphing Reciprocals of Linear Functions 8. Absolute Value and Reciprocal Functions 8. The graph of the absolute value function resembles a letter V. This video lecture walks through 10 examples on solving absolute value problems with a step by step approach. • generalizing a rule for writing absolute value functions in piecewise notation. Points on the graph whose y-coordinates are positive or equal to zero, (above and including the x-axis) will remain the same. Definition. Trigonometric Equations. 7 absolute value functions & reciprocal functions. The minimum value of -0. Chapter 7 : Absolute Value & Reciprocal Functions Calculus 12 Course Review. Absolute Value and Reciprocal Function Review - March 15 their preparation for the Test. Practice Test 8. It does not give you any math credits towards a degree. cite and apply the reciprocal, Pythagorean and cofunction identities. This lesson combines the ideas of absolute value, order of operations, operations with positives and negatives, and operations with integers and fractions. Precalculus Review: Lesson 1 In this video lesson we review how to graph functions using transformation techniques by walking through countless examples of different types of functions. We spent most of our time in that section looking at functions graphically because they were, after all, just sets of points in the plane. Pre-Calculus 11 and 12 is designed for students who intend to pursue a career in mathematics, sciences, or engineering. As the denominator of a rational function goes to 0 (and assuming the numerator is not also 0), the fraction becomes very large. Suderman's Math Website: Menu Pre-Calculus 11 Foundations of Math 11 Pre-Calculus 12 Ch. When you put a value for the domain in the function, you will get the resulting value in the range. of calculus. May 11th: Test Absolute Value & Reciprocal Functions May 14th: Graphing Reciprocals of Linear Functions May 15th: Graphing Reciprocals of Quadratic Functions May 16th: work period May 17th: Absolute Value of a Real Number May 18th: Sketching Absolute Value Functions May 22nd: Piecewise Notation May 23rd: Solving Absolute Value Equations May. f of x equals x cubed, the cubing function. CC+ College Algebra. 6 introduced several non-mathematical functions specific to the computer floating-point representation of numbers. Where is the singularity of this function? At x = 0. Focus On Chapter 7 7. Graph functions using stretches and compressions Perform a sequence of transformations 3. The domain of a rational function is all real numbers except the zeros of D(x). We look at a number of examples of circle and semi-circle functions, sketch their graphs, work out their domains and ranges, determine the centre and radius of a circle given its function, etc. 375 #1-14 7. absolute_value___reciprocal_functions_practice_test_answers. f of x equals square root x. The high school math courses emphasize higher order thinking skills, and use practical, real-life math examples to teach the material. quantities involving functions and their inverses RFR. 1 Create a table of values for yfx= (), given a table of values for y=fx( ). Section (Assignment) Absolute Value and Reciprocal Functions TEST. Even though you're involved with pre-calculus, you remember your old love, algebra, and that fact that absolute values then usually had two possible solutions. B: Protect the second input of the rem function from going to zero. A = 1 2 bh = 1 2 | |30−0 ( )5 = 75 The triangle has an area of 75 square units.