Chapter 10: Topics in Analytic Geometry

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 10: Topics in Analytic Geometry"

Transcription

1 Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed line ( d, the directrix). Note FP = TP. The axis or axis of symmetry of the parabola above is the y-axis. The focus is always on this axis and it is always perpendicular to the directrix. The segment through F and perpendicular to the axis is the latus rectum or focal width of the parabola. The point of the parabola midway between focus and directrix is called the vertex (V). If we take the vertex of the parabola to be at the origin (0, 0), the axis vertical as above, the focus at F(0, p), the directrix y = p, and focal width = 4p, we find the equation of the parabola: x = 4 p ythe parabola opens up (as shown) if p > 0, and down if p < 0. If we take the vertex of the parabola to be at the origin (0, 0), the axis horizontal, the focus at F(p, 0), the directrix x = p, and focal width = 4p, we find the equation of the parabola: y = 4 p xthe parabola opens right if p > 0, and left if p < 0. 1

2 Find vertex, focus, directrix, symmetry axis, and latus rectum of x + 6y = 0. The equation is of the form x = 4px if we write x = 6y. Since p = 3/ < 0, this parabola opens downward. The vertex is at V(0, 0), the focus F(0, 3/), the directrix is the horizontal line y = 3/, the symmetry axis is the y-axis (x = 0), and the latus rectum or focal with is 6. The parabola is graphed below Find an equation for the parabola that has its vertex at the origin and opens upward with focus 5 units from vertex. The focus is p = 5 from vertex and the equation is: x = 0y Find an equation for the parabola that has its vertex at the origin and directrix x =. The directrix is vertical so the parabola is horizontal, opening away from the directrix or to the left. The distance from V to directrix is p so p = and the equation is: y = 8x

3 Find vertex, focus, directrix, symmetry axis, and latus rectum of y 4x = 0. Writing y = 4x and comparing to y = 4px, we see the vertex is at V(0, 0) and p = 1. From the form of the equation and positive p we note the parabola opens to the right. The focus (always inside the parabola) is F(1, 0), the directrix is the vertical line x = 1, the symmetry axis is the x-axis (y = 0), and the latus rectum or focal width is 4p =

4 10. Ellipses The ellipse is the set of all points in the plane the sum of whose distances from two fixed points (the foci) is a constant (a). Let the center of the horizontal ellipse above be at (0, 0). The major axis (long axis) of the ellipse is then the horizontal segment stretching through the foci from one side of the ellipse to the other. The major axis intersects the ellipse at the vertices (V). The length of the major axis is a. The minor axis is the shorter axis of the ellipse, passing through the center and perpendicular to the major axis. It intersects the ellipse at the minor axis end-points B. The length of the minor axis is b. Each focus is a distance c (linear eccentricity) from the center. The foci are on the major axis. c = a b for the ellipse. The latus rectum or focal width of the ellipse is the segment through each focus perpendicular to the major axis. Its length is b / a. The eccentricity of the ellipse is a measure of its deviation from circularity. Always for the ellipse 0 < e < 1, with e = c/a. As e 0, the ellipse is more like a circle; as e 1, the ellipse is elongated. If the ellipse has center (0, 0), is longer in the horizontal direction, has vertices V(±a, 0), endpoints of minor axis B(0, ±b), foci F(±c, 0), and latus rectum endpoints (±c, ±b /a): x a y + = 1 is the equation of the horizontal ellipse b If the ellipse has center (0, 0), is longer in the vertical direction, has vertices V(0, ±a), endpoints of minor axis B(±b, 0), foci F(0, ±c), and latus rectum endpoints (±b /a, ±c): y a x + = 1 is the equation of the vertical ellipse. b 4

5 An alternative definition for the ellipse may be given in terms of directrices, pictured above. The ellipse is the set of points in the plane such that the distance of a point from a focus is e times the distance of the point from the associated directrix. The directrices above have equations x = ±a/e. Find V, B, F, e, length of major and minor axes, length of latera recta, endpoints of latera recta, and graph the ellipse below. x y + = 1 The form of this equation tells us the ellipse is horizontal, center origin. 5 9 a = 5, b = 9 (a is always the larger; if under x, the ellipse is horizontal) c = a b = 16 Thus a = 5, b = 3, c = 4. V(±5, 0) are endpoints of major axis. Major axis has length a = 10. B(0, ±3) are endpoints of minor axis. Minor axis has length b = 6. F(±4, 0) are coordinates of foci. They are always on major axis. e = c/a = 4/5 or 0.8 an elongated ellipse. Each latus rectum has length b /a = 18/5. This is the focal width of the ellipse. Endpoints of latera recta at (±4, ±9/5)

6 Find the equation of the ellipse with endpoints of major axis (±10, 0), distance between foci 6. The center is midway between V, B, F so here it is (0, 0). The major axis is horizontal, of length a = 10, so a = 5. The distance between foci is c = 6 so c = 3. Since c = a b, we find b = 5 3 = 16 and b = 4. Thus the equation: x y + = Find the equations of the ellipse with eccentricity 1/9, foci (0, ±). We see the center, midway between the foci, is at (0, 0). Since the foci are on the y-axis the ellipse is vertical. The distance from center to focus = c =. Now e = c/a 1/9 = /a a = 18. As above we use b = 18 = 30 and the desired equation is: y x + = Given 9x + 4y = 36, find: V, B, F, e, a, b, lengths and endpoints of latera recta, and graph The equation may be rewritten as: y x + = We see from its form that the center is (0, 0) and the ellipse is vertical. a = 9 and b = 4 c = 9 4 = 5 and a = 3, b =, and c = 5. Thus V(0, ±3), B(±, 0), F(0, ± 5), e = 5/3, length latus rectum = 8/5, and endpoints latera recta = (±4/5, ± 5)

7 10.3 Hyperbolas Hyperbolas may be defined as the set of points in the plane the difference of whose distances from two fixed points (the foci) is a constant (a). Alternatively, the distance of every point from a focus is e (the eccentricity) times greater than its distance from a fixed line associated with the focus called the directrix. We consider hyperbolas with center (0, 0). If the two branches of this curve open left and right (as pictured), then the distance between the vertices (transverse axis, VV') is a. Unlike the ellipse, this need not be the longer axis. The length of the conjugate axis BB' is b. The B points are not on the hyperbola, but help to define the rectangle shown above. The diagonals of the rectangle are the asymptotes, lines the hyperbola approach ever more closely as we get far from the origin. The foci are each c from the center, and located on the transverse axis. c = a + b The width of each branch of the hyperbola, measured through each focus, is the latus rectum or focal width. Each latus rectum has length b /a. The eccentricity e = c/a >1 fro the hyperbola. The asymptotes have slope ±b/a if the hyperbola opens left and right, ±a/b if it opens up and down. The equations of the asymptotes are y = (slope)x when hyperbola center at (0, 0). If the hyperbola has center (0, 0), opens left and right, has endpoints of transverse axis V(±a, 0), endpoints of conjugate axis B(0, ±b), foci F(±c, 0), latera recta endpoints (±c, ±b /a), and asymptotes y = ±(b/a)x, then it has equation: x y = 1 a b If the hyperbola has center (0, 0), opens up and down, has endpoints of transverse axis V(0, ±a), endpoints of conjugate axis B(±b, 0), foci F(0, ±c), latera recta endpoints (±b /a, ±c), and asymptotes y = ±(a/b)x, then it has equation: y x = 1 Note the + term is a. It may or may not be larger than b a b 7

8 Find V, B, F, e, length of transverse and conjugate axes, lengths of latera recta and their endpoints, equations of the asymptotes, and graph the hyperbola below. y x = 1 The equation implies center (0, 0), opens up and down 9 5 a = 9 (the + term - under y so a vertical hyperbola), b = 5, c = a + b = 34 Thus a = 3, b = 5, c = 34 Thus V(0, ±3), B(±5, 0), F(0, ± 34), e = ( 34)/3, transverse = 6, conjugate = 10, latera recta = 50/3, endpoints of latera recta (±5/3, ± 34), asy y = ±(3/5)x Write the equation for the hyperbola with foci (±5, 0), transverse axis = 6 The foci on the x-axis tell us the hyperbola opens left and right. The center is midway between the foci and is (0, 0). The foci are c from the center so c = 5. The length of the transverse axis is a = 6, so a = 3. Since c = a + b, we find b = 5 3 = 16. The desired equation is: x y =

9 10.4 Shifted Conics The center of parabolas, ellipses, and hyperbolas may not be at the origin. If the center is shifted horizontally and/or vertically to (h, k) then we replace x x h, and y y k. All features are measured from the center. Parabola: vertex (h, k), horizontal ( y k) = 4 p ( x h) Parabola vertex (h, k), vertical ( ) ( x h = 4 p y k) Ellipse center (h, k) horizontal Ellipse center (h, k) vertical Hyperbola center (h, k) horizontal Hyperbola center (h, k) vertical ( x h) ( y k) a + = 1 b ( y k) ( x h) a + = 1 b ( x h) ( y k) a = 1 b ( y k) ( x h) a = 1 b General Equation of a Shifted Conic The graph of the equation Ax + Cy + Dx + Ey + F = 0 where A and C are not both 0, is a conic or a degenerate conic. In the non-degenerate cases, the graph is: 1. Parabola is A or C is 0. Ellipse if A and C have same sign (or circle if A = C) 3. Hyperbola if A and C have opposite signs. Identifying Conics by the Discriminant (B 4 A C) The graph of the equation Ax + Bxy + Cy + Dx + Ey + F = 0, is a conic or a degenerate conic. In the non-degenerate cases, the graph is: 1. Parabola if B 4 A C = 0. Ellipse if B 4 A C < 0 3. Hyperbola if B 4 A C > 0 Find the equation of the hyperbola with one asymptote having equation y = x + 1, and vertices (ends of transverse axis) at (0, 0) and (0, ). The vertices are on a vertical transverse axis, so the hyperbola opens up and down. The center is midway between the vertices at (0, 1). The length of the transverse axis is = a so a = 1. The slope of the asymptote (a/b) = 1, so b = 1. ( ) ( ) ( ) y 1 x 0 y 1 x The equation is: = 1 =

10 Complete the square, determine the type of conic, find coordinates and lengths for all features, and graph 16x 9y 96x = 0. ( ) ( ) ( ) ( ) ( x 3) ( y 0) 16 x 6x + 9 y 0 = 0 16 x 6x y 0 = 144 = This is a hyperbola opening left and right, with center (3, 0) a = 9, b = 16, c = = 5 a = 3, b = 4, c = 5. V(0, 0)(6, 0) B(3, 4)(3, 4) F(, 0)(8, 0) e = 5/3 transverse axis length = 6, conjugate axis length = 8, latera recta length = 3/3 ends of latera recta(, ±16/3)(8, ±16/3), asy y 0 = ±(4/3)(x 3) Find vertex, focus, directrix, symmetry axis, focal width, and endpoints of latus rectum for the parabola: (x 3) = 8(y + 1) x vertical, vertex (3, 1), p> 0 so opens up, 4p = 8, so p =. Focus inside parabola so up from vertex or focus (3, 1). Symmetry axis is x = 3, directrix is y = 3. Focal width = 8 so ends of latus rectum at ( 1, 1) and (7, 1). 10

11 10.5 Rotation of Axes Given the most general form of second-degree equation below, reduce it to a simpler form by rotation of axes in order to find a conic horizontal or vertical relative to the new axes. Then use the results of previous sections to find components and graph. The most general form of second-degree equation is: Ax + Bxy + Cy +Dx + Ey + F = 0. We seek to find a positive, acute angle θ that will eliminate the xy term, giving: A'x' + C'y' +D'x' + E'y' + F' = 0, where the B' coefficient has been made zero. To do this, choose θ such that tan( θ) = B / (A C) if A C (else use θ = 45 ). Now replace every x and y in the original equation with: x = x' cos(θ) y' sin(θ) and y = x' sin(θ) + y' cos(θ) Given: 5x + 7xy +73y = 40x 30y + 75 Rotate by angle θ to eliminate x'y' term in new x'-y' coordinate system. Choose tan(θ) = B / (A C) = 7 / (5 73) = 7 / 1 = 4 / 7 Draw θ in quadrant I if positive, II if negative since θ is in interval (0, 180 ) We found the hypotenuse of 5 using the Pythagorean theorem. Observe cos(θ) = 7 / 5 and use the identities below. cos(θ) = 1+ cos(θ) = 3 / 5 sin(θ) = 1 cos(θ) = 4 / 5 Thus θ is a bit more than

12 Replace every x and y in the original expression with: x = x' cos(θ) y' sin(θ) and y = x' sin(θ) + y' cos(θ) x = x' y' and y = x' + y' Substituting, ( x' y' ) + 7( x' y' )( x' + y' ) + 73( x' + y' ) = ( x' y' ) 30( x' + y' ) Expanding, 1 [5(9x' 4x'y' + 16y' ) + 7(1x' 7x'y' 1y' ) + 73(16x' + 4x'y' + 9y' )] = 5 4x' 3y' 4x' 18y' + 75 Collecting like terms, x' ( ) + x'y'( ) + y' ( ) = 50y' Simplifying, 100x' + 5y' = 50y' + 75 or 4x' + y' = y' + 3 Translating (complete the square), 4x' + ( y' + y' + ) = 3 + or 4x' + (y' + 1) = 4 Writing in standard form, ( x' 0) ( y' + 1) = 1 An ellipse with center (0, 1) 1

13 Graphing, Maple plot: implicitplot(5*x^ + 7*x*y + 73*y^ = 40*x - 30*y + 75,x=-5..5,y=-5..5); Note: If one simply wishes to rotate axes by a given angle and transform single points to the new system, the inverse of the above transformations may be used. These are: x' = x cos(θ) + y sin(θ) and y' = x sin(θ) + y cos(θ) 13

14 10.6 Polar Equations of Conics ed ed r = or r = 1± e cos θ 1± e sin θ ( ) ( ) is a conic with one focus at origin and eccentricity e. The conic is a parabola is e < 1, an ellipse if 0 < e < 1, and a hyperbola if e > 1. x = 1 1 Pictured above is a graph of r =. This has e = 1 and d = cos( θ ) The focus of the parabola is at the pole, its vertex at (1/, 0 ). The directrix is to the right of the pole at x = 1 when using + in denominator. 1 If the equation were r =, the parabola would open to the right. 1 cos( θ ) The directrix would be x = 1. The focus would still be at the pole, the vertex now at (1/, 180 ). 1 If the equation were r =, the parabola would open down. 1 + sin( θ ) The directrix would be y = 1. (Note + directrix when + in denominator) The focus would still be at the pole, the vertex now at (1/, 90 ). 1 If the equation were r =, the parabola would open up. 1 sin( θ ) The directrix would be y = 1. The focus would still be at the pole, the vertex now at (1/, 70 ). 14

15 Example: Identify the conic 6 r = + sin θ We must put this in the form ( ) ed r = so divide num and denom by 1 + e sin θ 3 r = e = 1/ and d = 6. Since e = 1/, this is an ellipse sin( θ) Since there is a +sin(θ) in denominator, the ellipse is vertical, with its upper focus at the pole. To graph quickly, find r when θ = 0, 90, 180, and 70. ( )

16 10.7 Parametric Equations It is often useful to express x and y coordinates in terms of a parameter such as time t. We write x = f(t) and y = g(t) as the parametric equations of a curve. Sketch the curve defined by x = t, y = t, for t 4. One prepares a table of values of t and the corresponding x and y. Only the (x, y) values are plotted t x y Given parametric equations x = t, y = t, for t 4, eliminate the parameter to obtain a single equation in x and y. Solve for t in one of the equations and then substitute into the other. t = y x = (y ) or (y ) = x Recall (y k) = 4 p (x h) is the standard form of a shifted parabola that opens to the right with vertex at (h, k). Thus the graph should have vertex at (0, ) and open right. However, we must note t lies between and 4 meaning the curve and graph are only defined for 4 x 16 and 0 y. We draw this only. Note: To graph parametric equations with Mathematica, use ParametricPlot[ ]. On the TI-86, from the Home screen set Mode Param. Given x = sin(t) and y = 3 cos(t), eliminate the parameter. Square both equations: x = 4 sin (t) and y = 9 cos (t) or x y = sin () t and = cos () t Now add these equations. 4 9 x y + = sin () t + cos () t = 1 Vertical ellipse, center (0, 0)

17 Given x = sin(3 t) and y = sin(4 t), graph this Lissajous figure Note: Projectile motion equations are conveniently written in parametric form with time t as the parameter. If the initial velocity of a projectile is v 0 and it is launched at an angle θ above the horizontal, then we can resolve v 0 into vertical (v 0 sin(θ)) and horizontal (v 0 cos(θ)) components. In the absence of air resistance, and with gravity providing an acceleration (g = 3 ft/s = 9.80 m/s near the surface of the earth, directed downward) only in the y direction, we write x = x 0 + (v 0 cos(θ)) t and y = y 0 + (v 0 sin(θ)) t (1/) g t (x 0, y 0 ) is the starting position of the projectile. It is often taken to be (0, 0). Suppose a projectile is fired from position (0, 0) at t = 0 with an initial speed of 048 ft/s at an angle of 30 above the horizontal. What is the maximum height attained by the projectile? When and where will the projectile hit the ground? Graph. Eliminate the parameter t and show the path is a parabola. First we solve for the time the projectile is in the air. Assuming level ground, we find at what times the projectile is at y = 0: Solving 0 = 0 + (048 sin(30 )) t (1/) 3 t we find t = 0 and 64 s. The time up equals the time down for level ground so t = 3 s to max height. Maximum height = 0 + (048 sin(30 )) 3 (1/) 3 3 = 16,384 ft. Hits ground at t = 64 s and x = 0 + (048 cos(30 )) 64 = 113,51 ft 17

18 Now we must eliminate the parameter t and show the path is a parabola. Assume (x 0, y 0 ) is (0, 0). Solve for t in the x equation below and substitute. Thus x = (v 0 cos(θ)) t and y = (v 0 sin(θ)) t (1/) g t becomes: x x 1 x t = y= v0 sin( θ) g v0 cos( θ) v0 cos( θ) v0 cos( θ) We stop here as we have an equation of the form y = c 1 x c x, a parabola Note: Note: It is shown in the text that the range downfield for level ground may be written: v0 sin( θ) x = This agrees with the value for maximum x found above. g Polar equations may be put into parametric form. Recall: x = r cos(θ) and y = r sin(θ). Then if one is give r = f(θ), write: x = f(θ) cos(θ) and y = f(θ) sin(θ) Write r = sin(4θ) in parametric form. x = sin 4θ cos θ y = sin 4θ sin θ ( ) ( ) ( ) ( ) 18

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

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

Section 10.7 Parametric Equations

Section 10.7 Parametric Equations 299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x- (rcos(θ), rsin(θ)) and y-coordinates on a circle of radius r as a function of

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1 Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

More information

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved. 2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results

More information

Unit 9: Conic Sections Name Per. Test Part 1

Unit 9: Conic Sections Name Per. Test Part 1 Unit 9: Conic Sections Name Per 1/6 HOLIDAY 1/7 General Vocab Intro to Conics Circles 1/8-9 More Circles Ellipses 1/10 Hyperbolas (*)Pre AP Only 1/13 Parabolas HW: Part 4 HW: Part 1 1/14 Identifying conics

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

More information

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4. MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

More information

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

More information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

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

Section 10.5 Rotation of Axes; General Form of a Conic

Section 10.5 Rotation of Axes; General Form of a Conic Section 10.5 Rotation of Axes; General Form of a Conic 8 Objective 1: Identifying a Non-rotated Conic. The graph of the equation Ax + Bxy + Cy + Dx + Ey + F = 0 where A, B, and C cannot all be zero is

More information

Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 2015 Solutions for Class Activity #2 Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

Functions and Equations

Functions and Equations Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

More information

Example 1. Example 1 Plot the points whose polar coordinates are given by

Example 1. Example 1 Plot the points whose polar coordinates are given by Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

More information

EER#21- Graph parabolas and circles whose equations are given in general form by completing the square.

EER#21- Graph parabolas and circles whose equations are given in general form by completing the square. EER#1- Graph parabolas and circles whose equations are given in general form by completing the square. Circles A circle is a set of points that are equidistant from a fixed point. The distance is called

More information

Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2

Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2 1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect

More information

Engineering Geometry

Engineering Geometry Engineering Geometry Objectives Describe the importance of engineering geometry in design process. Describe coordinate geometry and coordinate systems and apply them to CAD. Review the right-hand rule.

More information

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s

Determine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,

More information

Parametric Equations and the Parabola (Extension 1)

Parametric Equations and the Parabola (Extension 1) Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when

More information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space 11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

More information

New York State Student Learning Objective: Regents Geometry

New York State Student Learning Objective: Regents Geometry New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students

More information

Solving Quadratic Equations

Solving Quadratic Equations 9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

5.1 Vector and Scalar Quantities. A vector quantity includes both magnitude and direction, but a scalar quantity includes only magnitude.

5.1 Vector and Scalar Quantities. A vector quantity includes both magnitude and direction, but a scalar quantity includes only magnitude. Projectile motion can be described by the horizontal ontal and vertical components of motion. In the previous chapter we studied simple straight-line motion linear motion. Now we extend these ideas to

More information

14. GEOMETRY AND COORDINATES

14. GEOMETRY AND COORDINATES 14. GEOMETRY AND COORDINATES We define. Given we say that the x-coordinate is while the y-coordinate is. We can view the coordinates as mappings from to : Coordinates take in a point in the plane and output

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

TOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM

TOMS RIVER REGIONAL SCHOOLS MATHEMATICS CURRICULUM Content Area: Mathematics Course Title: Precalculus Grade Level: High School Right Triangle Trig and Laws 3-4 weeks Trigonometry 3 weeks Graphs of Trig Functions 3-4 weeks Analytic Trigonometry 5-6 weeks

More information

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

More information

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points? GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

More information

Calculus with Analytic Geometry I Exam 10 Take Home part

Calculus with Analytic Geometry I Exam 10 Take Home part Calculus with Analytic Geometry I Exam 10 Take Home part Textbook, Section 47, Exercises #22, 30, 32, 38, 48, 56, 70, 76 1 # 22) Find, correct to two decimal places, the coordinates of the point on the

More information

Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

Algebra II and Trigonometry

Algebra II and Trigonometry Algebra II and Trigonometry Textbooks: Algebra 2: California Publisher: McDougal Li@ell/Houghton Mifflin (2006 EdiHon) ISBN- 13: 978-0618811816 Course descriphon: Algebra II complements and expands the

More information

3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models

3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.1 Quadratic Equations and Models Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.1-1 Polynomial Function A polynomial function of degree n, where n

More information

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

More information

Geometry Enduring Understandings Students will understand 1. that all circles are similar.

Geometry Enduring Understandings Students will understand 1. that all circles are similar. High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

More information

ME 111: Engineering Drawing

ME 111: Engineering Drawing ME 111: Engineering Drawing Lecture 4 08-08-2011 Engineering Curves and Theory of Projection Indian Institute of Technology Guwahati Guwahati 781039 Eccentrici ty = Distance of the point from the focus

More information

POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR COORDINATES DEFINITION OF POLAR COORDINATES POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand

More information

MATH 21. College Algebra 1 Lecture Notes

MATH 21. College Algebra 1 Lecture Notes MATH 21 College Algebra 1 Lecture Notes MATH 21 3.6 Factoring Review College Algebra 1 Factoring and Foiling 1. (a + b) 2 = a 2 + 2ab + b 2. 2. (a b) 2 = a 2 2ab + b 2. 3. (a + b)(a b) = a 2 b 2. 4. (a

More information

String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies

String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies String Art Mathematics: An Introduction to Geometry Expressions and Math Illustrations Stephen Arnold Compass Learning Technologies Introduction How do you create string art on a computer? In this lesson

More information

Chapter 8. Quadratic Equations and Functions

Chapter 8. Quadratic Equations and Functions Chapter 8. Quadratic Equations and Functions 8.1. Solve Quadratic Equations KYOTE Standards: CR 0; CA 11 In this section, we discuss solving quadratic equations by factoring, by using the square root property

More information

4.1 Radian and Degree Measure

4.1 Radian and Degree Measure Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

More information

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

More information

Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

More information

Math Review Large Print (18 point) Edition Chapter 2: Algebra

Math Review Large Print (18 point) Edition Chapter 2: Algebra GRADUATE RECORD EXAMINATIONS Math Review Large Print (18 point) Edition Chapter : Algebra Copyright 010 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRADUATE RECORD EXAMINATIONS,

More information

3. Double Integrals 3A. Double Integrals in Rectangular Coordinates

3. Double Integrals 3A. Double Integrals in Rectangular Coordinates 3. Double Integrals 3A. Double Integrals in ectangular Coordinates 3A-1 Evaluate each of the following iterated integrals: c) 2 1 1 1 x 2 (6x 2 +2y)dydx b) x 2x 2 ydydx d) π/2 π 1 u (usint+tcosu)dtdu u2

More information

Introduction to Vectors

Introduction to Vectors Introduction to Vectors A vector is a physical quantity that has both magnitude and direction. An example is a plane flying NE at 200 km/hr. This vector is written as 200 Km/hr at 45. Another example is

More information

Advanced Math Study Guide

Advanced Math Study Guide Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular

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

Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular. CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

More information

Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11}

Mathematics Pre-Test Sample Questions A. { 11, 7} B. { 7,0,7} C. { 7, 7} D. { 11, 11} Mathematics Pre-Test Sample Questions 1. Which of the following sets is closed under division? I. {½, 1,, 4} II. {-1, 1} III. {-1, 0, 1} A. I only B. II only C. III only D. I and II. Which of the following

More information

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form

ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola

More information

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point. 6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.

More information

Solutions to Exercises, Section 5.1

Solutions to Exercises, Section 5.1 Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

More information

Conjectures. Chapter 2. Chapter 3

Conjectures. Chapter 2. Chapter 3 Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

More information

Two vectors are equal if they have the same length and direction. They do not

Two vectors are equal if they have the same length and direction. They do not Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

More information

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

More information

Astromechanics Two-Body Problem (Cont)

Astromechanics Two-Body Problem (Cont) 5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

More information

Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Trigonometry Lesson Objectives

Trigonometry Lesson Objectives Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

More information

Vectors; 2-D Motion. Part I. Multiple Choice. 1. v

Vectors; 2-D Motion. Part I. Multiple Choice. 1. v This test covers vectors using both polar coordinates and i-j notation, radial and tangential acceleration, and two-dimensional motion including projectiles. Part I. Multiple Choice 1. v h x In a lab experiment,

More information

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

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

More information

Solutions to Homework 10

Solutions to Homework 10 Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

More information

1.7 Cylindrical and Spherical Coordinates

1.7 Cylindrical and Spherical Coordinates 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

Understanding Basic Calculus

Understanding Basic Calculus Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

More information

MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

More information

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu) 6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

More information

MODERN APPLICATIONS OF PYTHAGORAS S THEOREM

MODERN APPLICATIONS OF PYTHAGORAS S THEOREM UNIT SIX MODERN APPLICATIONS OF PYTHAGORAS S THEOREM Coordinate Systems 124 Distance Formula 127 Midpoint Formula 131 SUMMARY 134 Exercises 135 UNIT SIX: 124 COORDINATE GEOMETRY Geometry, as presented

More information

TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

TWO-DIMENSIONAL TRANSFORMATION

TWO-DIMENSIONAL TRANSFORMATION CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

More information

Additional Topics in Math

Additional Topics in Math Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

More information

Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Exam #3 Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

More information

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

More information

SAT Subject Math Level 2 Facts & Formulas

SAT Subject Math Level 2 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses

More information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

More information

REVIEW OF CONIC SECTIONS

REVIEW OF CONIC SECTIONS REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result

More information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

7.3 Volumes Calculus

7.3 Volumes Calculus 7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are

More information

Introduction to Calculus

Introduction to Calculus Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative

More information

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145: MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

More information

Mark Howell Gonzaga High School, Washington, D.C.

Mark Howell Gonzaga High School, Washington, D.C. Be Prepared for the Calculus Exam Mark Howell Gonzaga High School, Washington, D.C. Martha Montgomery Fremont City Schools, Fremont, Ohio Practice exam contributors: Benita Albert Oak Ridge High School,

More information

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}.

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}. Use a table of values to graph each equation. State the domain and range. 1. y = x 2 + 3x + 1 x y 3 1 2 1 1 1 0 1 1 5 2 11 Graph the ordered pairs, and connect them to create a smooth curve. The parabola

More information

7.3 Parabolas. 7.3 Parabolas 505

7.3 Parabolas. 7.3 Parabolas 505 7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information