Sample Problems. Practice Problems

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Sample Problems. Practice Problems"

Transcription

1 Lecture Notes Circles - Part page Sample Problems. Find an equation for the circle centered at (; ) with radius r = units.. Graph the equation + + = ( ).. Consider the circle ( ) + ( + ) =. Find all points on the circle with coordinate being.. Find the coordinates of all points where the circle + ( ) = and the line = + 8. a) Consider the circle + =. Find an equation for the tangent line drawn to the circle at the point (; ). b) Consider the circle + + =. Find an equation for the tangent line drawn to the circle at the point (; ).. a) Find the points where the circles ( + ) + ( + ) = and ( + ) + ( ) = b) Find the points where the circles ( + ) + ( + ) = and ( ) + ( ) = Practice Problems. Find an equation for each of the following circles. C will denote the center and r the radius. a) C (; ) r = 7 b) C (; ) r = c) C ( 8; ) r = p. Graph the equation + + = ( + ). Consider the circle ( + 7) + ( ) =. Find all points on the circle a) with coordinate. c) with coordinate. b) with coordinate. d) with coordinate.. Find the coordinates of all point(s) where the circle and the line a) ( + ) + ( 7) = 8 and + = c) + ( + ) = and = b) ( ) + ( + ) = and = + 9 d) ( ) + ( + ) = and = +. Given the equation of a circle and a point P on it, nd an equation for the tangent line drawn to the circle at the point P. a) + = and P ( 8; ) b) ( ) + ( + ) = and P (; ). Find the coordinates of all points where the given circles a) ( ) + ( ) = and ( ) + ( ) = b) ( ) + = and + ( ) = c) + ( 8) = and ( 7) + ( ) = c copright Hidegkuti, Powell, 8 Last revised: March 9,

2 Lecture Notes Circles - Part page Sample Problems - Answers.) ( ) + ( + ) =.) ( + ) + ( ) =.) ( ; ) and (; ).) ( ; ) and (; ).) a) = + b) + = ( ) or =.) a) ( ; ) and ( ; ) b) ( ; ) and (; ) Practice Problems - Answers.) a) ( ) + = 9 b) + ( + ) = c) ( + 8) + ( ) =.) ( ) + ( + ) = center: (; ) radius: ) a) ( ; ) and ( ; ) b) ( 7; ) c) there is no such point d) ; p and ; + p.) a) ( ; 9) b) (; ) and (9; ) c) the don t intersect d) ( ; ) and (; ).) a) ( + 8) = or = + b) 7 ( ) = + or = 7 7.) a) (; ) b) (7; ) and (; ) c) the circles do not intersect c copright Hidegkuti, Powell, 8 Last revised: March 9,

3 Lecture Notes Circles - Part page Sample Problems - Solutions. Find an equation for the circle centered at (; ) with radius r = units. Solution: Let P (; ) be a general point on the circle. If the point P is on the circle, then it must be units awa from the point C (; ) : Consider the right triangle shown on the picture below. The sides of this triangle are j j, j + j, and : The Phtagorean theorem, stated for this triangle is ( ) + ( + ) = ; which is the equation for the circle.. Graph the equation + + = ( ) Solution: Complete the squares to read the center and radius. + + Thus the center is ( ; ) and the radius : + + = ( ) + + = + + = = ( + ) + ( ) = ( + ) + ( ) = c copright Hidegkuti, Powell, 8 Last revised: March 9,

4 Lecture Notes Circles - Part page. Consider the circle ( ) + ( + ) =. Find all points on the circle with coordinate being. Solution: We set = and solve for in the circle s equation. Because the equation is quadratic, we ma obtain two or one or no solution. It is also helpful to sketch a graph of the circle; it is centered at (; ) and has a radius of p. ( ) + ( + ) = ( ) = ( ) + ( ) = ( + ) ( ) = ( ) + = ( + ) ( ) = ( ) = = = Therefore, there are two points on this circle with coordinate ; and the are ( ; ) and (; ) Find the coordinates of all points where the circle + ( ) = and the line = + 8 Solution: We need to solve the following sstem of equations: + ( ) = = + 8 We will use substitution. We substitute = + 8 in the rst equation and solve for. + ( + 8 ) = = + ( + ) = = + + = ( ) ( + ) = + = = = We now use the second equation to nd the value belonging to the values we just obtained. If =, then = () + 8 = : If =, then = ( ) + 8 = : Thus the two points are (; ) and ( ; ). We check: both points should be on both the circle and the line. c copright Hidegkuti, Powell, 8 Last revised: March 9,

5 Lecture Notes Circles - Part page. Consider the circle + =. Find an equation for the tangent line drawn to the circle at the point (; ). Solution: The tangent line is perpendicular to the radius drawn to the point of tangenc. We can easil nd the slope of this radius as the slope of the line segment connecting the center of the circle, (; ) and the point of tangenc, (; ). We use the slope formula, slope = m = rise run = = = Since perpendicular to a line with slope, the tangent line must have slope, the negative reciprocal of. It must also pass through the point (; ) : The point-slope form of this line s equation is then = ( ) : We simplif this and obtain = b) Consider the circle + + =. Find an equation for the tangent line drawn to the circle at the point (; ). Solution: We rst transform the equation of the circle to determine its center s coordinates = + + = = ( + ) 9 + ( ) = ( + ) + ( ) = Thus the center of the circle is ( ; ) and its radius is c copright Hidegkuti, Powell, 8 Last revised: March 9,

6 Lecture Notes Circles - Part page The tangent line is perpendicular to the radius drawn to the point of tangenc. We can easil nd the slope of this radius: the slope of the line segment connecting the center ( ; ) and the point of tangenc (; ) is slope = m = rise run = = ( ) = Since perpendicular to a line with slope, the tangent line must have slope, the negative reciprocal of. The tangent line must also pass through the point (; ) : The point-slope form of this line s equation is then + = ( ). We can simplif this and obtain the slope-intercept form, = a) Find the points where the circles ( + ) + ( + ) = and ( + ) + ( ) = Solution: We need to solve the following sstem: ( + ) + ( + ) = ( + ) + ( ) = We multipl out the complete squares and combine like terms in both equations = = = = = = c copright Hidegkuti, Powell, 8 Last revised: March 9,

7 Lecture Notes Circles - Part page 7 We will multipl the second equation b and add the two equations. This will cancel out all quadratic terms. The sum of the two equations is = 7 + = + = divide both sides b + = = We substitute this into the rst equation and solve ( + ) + ( + ) = and = ( + ) + ( + ) = + A + ( + ) = ( + ) + ( + ) = = + = = ( ) = =) = = We now nd the values belonging to the values, using =. If =, then = = and if =, then = =. Thus the two circles intersect at the points ( ; ) and ( ; ) b) Find the points where the circles ( + ) + ( + ) = and ( ) + ( ) = Solution: We need to solve the following sstem: ( + ) + ( + ) = ( ) + ( ) = c copright Hidegkuti, Powell, 8 Last revised: March 9,

8 Lecture Notes Circles - Part page 8 We multipl out the complete squares and combine like terms in both equations = = = + = We will multipl the second equation b and add the two equations. This will cancel out all quadratic terms. The sum of the two equations is = + + = 8 + = divide both sides b + = We now solve for + = = + We substitute this into the rst equation and solve for. ( + ) + ( + ) = and = + ( + ) C {z A } = + 7 ( + ) + = ( + ) ( + 7) + = since + = 7 a b = a ( + ) ( + 7) + = multipl b ( + ) + ( + 7) = = = = = ( ) ( + ) = =) = = We now nd the values belonging to the values, using = () + = = and if =, then = intersect at the points ( ; ) and (; ) : ( ) + b +. If = ; then =. Thus the two circles For more documents like this, visit our page at and click on Lecture Notes. questions or comments to c copright Hidegkuti, Powell, 8 Last revised: March 9,

Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...

Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles... Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

More information

Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2 Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

More information

Sample Problems. Lecture Notes Equations with Parameters page 1

Sample Problems. Lecture Notes Equations with Parameters page 1 Lecture Notes Equations with Parameters page Sample Problems. In each of the parametric equations given, nd the value of the parameter m so that the equation has exactly one real solution. a) x + mx m

More information

Lesson 19: Equations for Tangent Lines to Circles

Lesson 19: Equations for Tangent Lines to Circles Classwork Opening Exercise A circle of radius 5 passes through points ( 3, 3) and (3, 1). a. What is the special name for segment? b. How many circles can be drawn that meet the given criteria? Explain

More information

Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown. Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

More information

2.3 Writing Equations of Lines

2.3 Writing Equations of Lines . Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

More information

Sample Problems. 6. The hypotenuse of a right triangle is 68 cm. The di erence between the other two sides is 28 cm. Find the sides of the triangle.

Sample Problems. 6. The hypotenuse of a right triangle is 68 cm. The di erence between the other two sides is 28 cm. Find the sides of the triangle. Lecture Notes The Pythagorean Theorem age 1 Samle Problems 1. Could the three line segments given below be the three sides of a right triangle? Exlain your answer. a) 6 cm; 10 cm; and 8 cm b) 7 ft, 15

More information

REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

More information

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form. Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

More information

-axis -axis. -axis. at point.

-axis -axis. -axis. at point. Chapter 5 Tangent Lines Sometimes, a concept can make a lot of sense to us visually, but when we try to do some explicit calculations we are quickly humbled We are going to illustrate this sort of thing

More information

Circle Name: Radius: Diameter: Chord: Secant:

Circle Name: Radius: Diameter: Chord: Secant: 12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

More information

Section 2.1 Rectangular Coordinate Systems

Section 2.1 Rectangular Coordinate Systems P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

More information

Lesson 19: Equations for Tangent Lines to Circles

Lesson 19: Equations for Tangent Lines to Circles Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

More information

21-114: Calculus for Architecture Homework #1 Solutions

21-114: Calculus for Architecture Homework #1 Solutions 21-114: Calculus for Architecture Homework #1 Solutions November 9, 2004 Mike Picollelli 1.1 #26. Find the domain of g(u) = u + 4 u. Solution: We solve this by considering the terms in the sum separately:

More information

5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope 5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

More information

Section 2.2 Equations of Lines

Section 2.2 Equations of Lines Section 2.2 Equations of Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes

More information

1 Lecture 19: Implicit differentiation

1 Lecture 19: Implicit differentiation Lecture 9: Implicit differentiation. Outline The technique of implicit differentiation Tangent lines to a circle Examples.2 Implicit differentiation Suppose we have two quantities or variables x and y

More information

The Parabola and the Circle

The Parabola and the Circle The Parabola and the Circle The following are several terms and definitions to aid in the understanding of parabolas. 1.) Parabola - A parabola is the set of all points (h, k) that are equidistant from

More information

Lesson 19: Equations for Tangent Lines to Circles

Lesson 19: Equations for Tangent Lines to Circles Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

More information

The Derivative and the Tangent Line Problem. The Tangent Line Problem

The Derivative and the Tangent Line Problem. The Tangent Line Problem The Derivative and the Tangent Line Problem Calculus grew out of four major problems that European mathematicians were working on during the seventeenth century. 1. The tangent line problem 2. The velocity

More information

Algebra. Indiana Standards 1 ST 6 WEEKS

Algebra. Indiana Standards 1 ST 6 WEEKS Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

More information

Lesson 6: Linear Functions and their Slope

Lesson 6: Linear Functions and their Slope Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation

More information

Analytic Geometry Section 2-6: Circles

Analytic Geometry Section 2-6: Circles Analytic Geometry Section 2-6: Circles Objective: To find equations of circles and to find the coordinates of any points where circles and lines meet. Page 81 Definition of a Circle A circle is the set

More information

Sample Problems. 3. Find the missing leg of the right triangle shown on the picture below.

Sample Problems. 3. Find the missing leg of the right triangle shown on the picture below. Lecture Notes The Pythagorean Theorem page 1 Sample Problems 1. Could the three line segments given below be the three sides of a right triangle? Explain your answer. a) 6 cm; 10 cm; and 8 cm b) 7 ft,

More information

Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines. The Slope of a Line Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

More information

Sample Problems. 2. Rationalize the denominator in each of the following expressions Simplify each of the following expressions.

Sample Problems. 2. Rationalize the denominator in each of the following expressions Simplify each of the following expressions. Lecture Notes Radical Exressions age Samle Problems. Simlify each of the following exressions. Assume that a reresents a ositive number. a) b) g) 7 + 7 c) h) 7 x y d) 0 + i) j) x x + e) k) ( x) ( + x)

More information

10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: 2. SOLUTION: 3.

10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: 2. SOLUTION: 3. Find x. Assume that segments that appear to be tangent are tangent. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. 5. SCIENCE A piece of broken pottery found at an archaeological site is shown.

More information

Pre-Calculus Review Problems Solutions

Pre-Calculus Review Problems Solutions MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry Pre-Calculus Review Problems Solutions Problem 1. Give equations for the following lines in both point-slope and slope-intercept form. (a) The

More information

10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: ANSWER: 2 SOLUTION: ANSWER:

10-7 Special Segments in a Circle. Find x. Assume that segments that appear to be tangent are tangent. 1. SOLUTION: ANSWER: 2 SOLUTION: ANSWER: Find x. Assume that segments that appear to be tangent are tangent. 1. 3. 2 5 2. 4. 6 13 esolutions Manual - Powered by Cognero Page 1 5. SCIENCE A piece of broken pottery found at an archaeological site

More information

Pre-Calculus III Linear Functions and Quadratic Functions

Pre-Calculus III Linear Functions and Quadratic Functions Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.3

More information

Linear Equations and Graphs

Linear Equations and Graphs 2.1-2.4 Linear Equations and Graphs Coordinate Plane Quadrants - The x-axis and y-axis form 4 "areas" known as quadrants. 1. I - The first quadrant has positive x and positive y points. 2. II - The second

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures

The measure of an arc is the measure of the central angle that intercepts it Therefore, the intercepted arc measures 8.1 Name (print first and last) Per Date: 3/24 due 3/25 8.1 Circles: Arcs and Central Angles Geometry Regents 2013-2014 Ms. Lomac SLO: I can use definitions & theorems about points, lines, and planes to

More information

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20 Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

5 $75 6 $90 7 $105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line?

5 $75 6 $90 7 $105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line? Review Slope & Equations of Lines Name Hour STANDARD FORM: Ax + By = C 1. What is the slope of a vertical line? 2. What is the slope of a horizontal line? 3. Is y = 4 the equation of a horizontal or vertical

More information

5.1 Writing Linear Equations in Slope-Intercept Form. 1. Use slope-intercept form to write an equation of a line.

5.1 Writing Linear Equations in Slope-Intercept Form. 1. Use slope-intercept form to write an equation of a line. 5.1 Writing Linear Equations in Slope-Intercept Form Objectives 1. Use slope-intercept form to write an equation of a line. 2. Model a real-life situation with a linear function. Key Terms Slope-Intercept

More information

Geometry SOL G.11 G.12 Circles Study Guide

Geometry SOL G.11 G.12 Circles Study Guide Geometry SOL G.11 G.1 Circles Study Guide Name Date Block Circles Review and Study Guide Things to Know Use your notes, homework, checkpoint, and other materials as well as flashcards at quizlet.com (http://quizlet.com/4776937/chapter-10-circles-flashcardsflash-cards/).

More information

Section 7.2 Linear Programming: The Graphical Method

Section 7.2 Linear Programming: The Graphical Method Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function

More information

Linear Equations Review

Linear Equations Review Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The y-intercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the y-intercept

More information

Level: High School: Geometry. Domain: Expressing Geometric Properties with Equations G-GPE

Level: High School: Geometry. Domain: Expressing Geometric Properties with Equations G-GPE 1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Translate between the geometric

More information

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button.

Reteaching Masters. To jump to a location in this book. 1. Click a bookmark on the left. To print a part of the book. 1. Click the Print button. Reteaching Masters To jump to a location in this book. Click a bookmark on the left. To print a part of the book. Click the Print button.. When the Print window opens, tpe in a range of pages to print.

More information

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2). Chapter Section : Equations of Lines Answers to Problems For problems -, put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find

More information

Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

More information

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System. Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

More information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its

More information

x 2 + y 2 = 25 and try to solve for y in terms of x, we get 2 new equations y = 25 x 2 and y = 25 x 2.

x 2 + y 2 = 25 and try to solve for y in terms of x, we get 2 new equations y = 25 x 2 and y = 25 x 2. Lecture : Implicit differentiation For more on the graphs of functions vs. the graphs of general equations see Graphs of Functions under Algebra/Precalculus Review on the class webpage. For more on graphing

More information

Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

More information

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin. 13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

More information

Solving Systems of Equations (Inequalities) with Parameters

Solving Systems of Equations (Inequalities) with Parameters Solving Systems of Equations (Inequalities) with Parameters Time required 90 5 minutes Teaching Goals:. Students apply a graphical method for solving systems of equations (inequalities) with parameters..

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Tangent line of a circle can be determined once the tangent point or the slope of the line is known.

Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : Ax + By

More information

Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line

Lines and planes in space (Sect. 12.5) Review: Lines on a plane. Lines in space (Today). Planes in space (Next class). Equation of a line Lines and planes in space (Sect. 2.5) Lines in space (Toda). Review: Lines on a plane. The equations of lines in space: Vector equation. arametric equation. Distance from a point to a line. lanes in space

More information

Sample Problems. Practice Problems

Sample Problems. Practice Problems Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these

More information

Sucesiones repaso Klasse 12 [86 marks]

Sucesiones repaso Klasse 12 [86 marks] Sucesiones repaso Klasse [8 marks] a. Consider an infinite geometric sequence with u = 40 and r =. (i) Find. u 4 (ii) Find the sum of the infinite sequence. (i) correct approach () e.g. u 4 = (40) u 4

More information

Section 3.4 The Slope Intercept Form: y = mx + b

Section 3.4 The Slope Intercept Form: y = mx + b Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear

More information

Slope-Intercept Equation. Example

Slope-Intercept Equation. Example 1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

More information

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

More information

Additional Examples of using the Elimination Method to Solve Systems of Equations

Additional Examples of using the Elimination Method to Solve Systems of Equations Additional Examples of using the Elimination Method to Solve Systems of Equations. Adjusting Coecients and Avoiding Fractions To use one equation to eliminate a variable, you multiply both sides of that

More information

Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a.

Section 7.1 Solving Linear Systems by Graphing. System of Linear Equations: Two or more equations in the same variables, also called a. Algebra 1 Chapter 7 Notes Name Section 7.1 Solving Linear Systems by Graphing System of Linear Equations: Two or more equations in the same variables, also called a. Solution of a System of Linear Equations:

More information

Study Guide and Review - Chapter 4

Study Guide and Review - Chapter 4 State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. The

More information

v x d ACP Algebra II Summer Review Packet ID: A Short Answer ;-5

v x d ACP Algebra II Summer Review Packet ID: A Short Answer ;-5 Class: Date: ACP Algebra II Summer Review Packet Short Answer Find the value of the given expression. I. t[32t(70(-21))] 2.1-90-;-5 3. Evaluate the given expression if w = 33, x = 1, y = 28, and z = 36.

More information

Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name:

Lyman Memorial High School. Pre-Calculus Prerequisite Packet. Name: Lyman Memorial High School Pre-Calculus Prerequisite Packet Name: Dear Pre-Calculus Students, Within this packet you will find mathematical concepts and skills covered in Algebra I, II and Geometry. These

More information

Lesson 1: Introducing Circles

Lesson 1: Introducing Circles IRLES N VOLUME Lesson 1: Introducing ircles ommon ore Georgia Performance Standards M9 12.G..1 M9 12.G..2 Essential Questions 1. Why are all circles similar? 2. What are the relationships among inscribed

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Mathematics 1. Lecture 5. Pattarawit Polpinit

Mathematics 1. Lecture 5. Pattarawit Polpinit Mathematics 1 Lecture 5 Pattarawit Polpinit Lecture Objective At the end of the lesson, the student is expected to be able to: familiarize with the use of Cartesian Coordinate System. determine the distance

More information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 2.2. Distance and Midpoint Formulas; Circles SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, :15 a.m. SAMPLE RESPONSE SET

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, :15 a.m. SAMPLE RESPONSE SET The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Question 29................... 2 Question 30...................

More information

Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems. Matthew Staley Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

More information

2.1 Equations of Lines

2.1 Equations of Lines Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

More information

Answers to Basic Algebra Review

Answers to Basic Algebra Review Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

More information

The Not-Formula Book for C1

The Not-Formula Book for C1 Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

More information

The Inscribed Angle Alternate A Tangent Angle

The Inscribed Angle Alternate A Tangent Angle Student Outcomes Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is

More information

Chapter 12. The Straight Line

Chapter 12. The Straight Line 302 Chapter 12 (Plane Analytic Geometry) 12.1 Introduction: Analytic- geometry was introduced by Rene Descartes (1596 1650) in his La Geometric published in 1637. Accordingly, after the name of its founder,

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

More information

MATH 111: EXAM 02 SOLUTIONS

MATH 111: EXAM 02 SOLUTIONS MATH 111: EXAM 02 SOLUTIONS BLAKE FARMAN UNIVERSITY OF SOUTH CAROLINA Answer the questions in the spaces provided on the question sheets and turn them in at the end of the class period Unless otherwise

More information

10.5 and 10.6 Lesson Plan

10.5 and 10.6 Lesson Plan Title: Secants, Tangents, and Angle Measures 10.5 and 10.6 Lesson Plan Course: Objectives: Reporting Categories: Related SOL: Vocabulary: Materials: Time Required: Geometry (Mainly 9 th and 10 th Grade)

More information

Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved.

1.3 LINEAR EQUATIONS IN TWO VARIABLES. Copyright Cengage Learning. All rights reserved. 1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

More information

Tools of Algebra. Solving Equations. Solving Inequalities. Dimensional Analysis and Probability. Scope and Sequence. Algebra I

Tools of Algebra. Solving Equations. Solving Inequalities. Dimensional Analysis and Probability. Scope and Sequence. Algebra I Scope and Sequence Algebra I Tools of Algebra CLE 3102.1.1, CFU 3102.1.10, CFU 3102.1.9, CFU 3102.2.1, CFU 3102.2.2, CFU 3102.2.7, CFU 3102.2.8, SPI 3102.1.3, SPI 3102.2.3, SPI 3102.4.1, 1-2 Using Variables,

More information

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

More information

Math 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas

Math 40 Chapter 3 Lecture Notes. Professor Miguel Ornelas Math 0 Chapter Lecture Notes Professor Miguel Ornelas M. Ornelas Math 0 Lecture Notes Section. Section. The Rectangular Coordinate Sstem Plot each ordered pair on a Rectangular Coordinate Sstem and name

More information

Higher Education Math Placement

Higher Education Math Placement Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

More information

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340 ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:

More information

Algebra 1-2. A. Identify and translate variables and expressions.

Algebra 1-2. A. Identify and translate variables and expressions. St. Mary's College High School Algebra 1-2 The Language of Algebra What is a variable? A. Identify and translate variables and expressions. The following apply to all the skills How is a variable used

More information

6.3 Polar Coordinates

6.3 Polar Coordinates 6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard

More information

In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

More information

Math 10 - Unit 7 Final Review - Coordinate Geometry

Math 10 - Unit 7 Final Review - Coordinate Geometry Class: Date: Math 10 - Unit Final Review - Coordinate Geometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the slope of this line segment.

More information

CK-12 Geometry: Parts of Circles and Tangent Lines

CK-12 Geometry: Parts of Circles and Tangent Lines CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.

More information

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304

Years t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304 Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations.

SECTION 0.11: SOLVING EQUATIONS. LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. (Section 0.11: Solving Equations) 0.11.1 SECTION 0.11: SOLVING EQUATIONS LEARNING OBJECTIVES Know how to solve linear, quadratic, rational, radical, and absolute value equations. PART A: DISCUSSION Much

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Pre-Algebra Curriculum Crawford Central School District

Pre-Algebra Curriculum Crawford Central School District Concept Competency Resources Larson Pre- Algebra Resource Vocabulary Strategy PA Core Eligible Content PA Core Standards PA Core Standards Scope and Sequence Number System identify numbers as either rational

More information

Unit 10: Quadratic Relations

Unit 10: Quadratic Relations Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

More information

Solving Systems of Equations Algebraically Examples

Solving Systems of Equations Algebraically Examples Solving Systems of Equations Algebraically Examples 1. Graphing a system of equations is a good way to determine their solution if the intersection is an integer. However, if the solution is not an integer,

More information

Systems of Equations Involving Circles and Lines

Systems of Equations Involving Circles and Lines Name: Systems of Equations Involving Circles and Lines Date: In this lesson, we will be solving two new types of Systems of Equations. Systems of Equations Involving a Circle and a Line Solving a system

More information

Algebra Course KUD. Green Highlight - Incorporate notation in class, with understanding that not tested on

Algebra Course KUD. Green Highlight - Incorporate notation in class, with understanding that not tested on Algebra Course KUD Yellow Highlight Need to address in Seminar Green Highlight - Incorporate notation in class, with understanding that not tested on Blue Highlight Be sure to teach in class Postive and

More information