THE PARABOLA section. Developing the Equation


 Jeremy Burke
 10 months ago
 Views:
Transcription
1 80 (0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or Minimum Value of The parabola is one of four different curves that can be obtained b intersecting a cone and a plane as in Fig... These curves, called conic sections, are the parabola, circle, ellipse, and hperbola. You studied parabolas in Section.6, but in this section ou will stud parabolas in more detail. You will see how the parabola and the other conic sections are used in applications such as satellite dishes, telescopes, spotlights, and navigation. Parabola Parabola Hperbola Focus Verte Circle Ellipse FIGURE. Developing the Equation FIGURE. Directri In Section.6 we called the graph of a b c a parabola. This equation is the standard equation of a parabola. In this section ou will see that the following geometric definition describes the same curve as the equation. Parabola Focus Given a line (the directri) and a point not on the line (the focus), the set of all points in the plane that are equidistant from the point and the line is called a parabola. FIGURE. p > 0 (0, p) (, ) (0, 0) = p (, p) FIGURE.6 The verte of the parabola is the midpoint of the line segment joining the focus and the directri, perpendicular to the directri. See Fig... The focus of a parabola is important in applications. When parallel ras of light travel into a parabolic reflector, the are reflected toward the focus as in Fig... This propert is used in telescopes to see the light from distant stars. If the light source is at the focus, as in a searchlight, the light is reflected off the parabola and projected outward in a narrow beam. This reflecting propert is also used in camera lenses, satellite dishes, and eavesdropping devices. To develop an equation for a parabola, given the focus and directri, choose the point (0, p), where p 0 as the focus and the line p as the directri, as shown in Fig..6. The verte of this parabola is (0, 0). For an arbitrar point (, ) on the parabola the distance to the directri is the distance from (, ) to(, p).
2 . The Parabola () 8 = p (, p) (0, 0) (, ) (0, p) p < 0 FIGURE.7 The distance to the focus is the distance between (, ) and (0, p). We use the fact that these distances are equal to write the equation of the parabola: ( 0) ( p) ( ) ( (p)) To simplif the equation, first remove the parentheses inside the radicals: p p p p p p p p p Square each side. p a Let a p. Subtract and p from each side. So a curve that satisfies the geometric definition of parabola has an equation of the form a b c. If the focus is at (0, p) with p 0 and the directri is p, then the parabola opens downward as shown in Fig..7. Deriving the equation from the, which is of the form. Note that a and p have the same sign because a. distances (as was just done for p 0) again ields p a for a p If a parabola has verte (0, 0) and opens up or down, then its equation is of the form a. In general, if (h, k) is the verte, (h, k p) is the focus, and k p is the directri, then we can develop the equation a( h) k for the parabola just as we developed the equation a. Graphs of a( h) k are shown in Fig..8. p a > 0 (h, k + p) (h, k) a p = a( h) + k Directri: = k p FIGURE.8 a < 0 Directri: = k p (h, k) = a( h) + k (h, k + p) Parabolas in the Form a( h) k The graph of the equation a( h) k (a 0) is a parabola with verte (h, k), focus (h, k p), and directri k p, where a. If a 0, the parabola opens upward; if a 0, the parabola opens downward. p Note that the locations of the focus and directri determine the value of a and the shape and opening of the parabola. CAUTION For a parabola that opens upward, p 0, and the focus (h, k p) is above the verte (h, k). For a parabola that opens downward, p 0, and the focus (h, k p) is below the verte (h, k). In either case the distance from the verte to the focus and the verte to the directri is p.
3 8 () Chapter Nonlinear Sstems and the Conic Sections E X A M P L E ( 0, = ( = FIGURE.9 Finding the verte, focus, and directri, given an equation Find the verte, focus, and directri for the parabola. Compare to the general formula a( h) k. We see that h 0, k 0, and a. So the verte is (0, 0). Because a, we can use a to p get, p or p. Use (h, k p) to get the focus 0,. Use the equation k p to get as the equation of the directri. See Fig..9. E X A M P L E ( 7, ( (, ) = Finding an equation, given a focus and directri Find the equation of the parabola with focus (, ) and directri. Because the verte is halfwa between the focus and directri, the verte is, 7. See Fig..0. The distance from the verte to the focus is. Because the focus is above the verte, p is positive. So p, and a. The equation is p ( ()) 7. Simplif to get the equation. FIGURE.0 The equation of a parabola can be written in two different forms. To change the form a( h) k to the form a b c, we square the binomial and combine like terms, as in Eample. To change from a b c to a( h) k, we complete the square. E X A M P L E Converting a b c to a( h) k Write in the form a( h) k and identif the verte, focus, and directri of the parabola. Use completing the square to rewrite the equation: ( ) ( ) Complete the square. ( ) ( ) Move () outside the parentheses. The verte is (, ). Because a, we have p, p
4 . The Parabola () 8 The graphs of and ( ) appear to be identical. This supports the conclusion that the equations are equivalent. calculator closeup 0 0 and p 8. Because the parabola opens upward, the focus is 8 unit above the verte at, 8, or, 8, and the directri is the horizontal line unit below the verte, or. 8 CAUTION Be careful when ou complete a square within parentheses as in Eample. For another eample, consider the equivalent equations ( ), ( ), and ( ). Identifing the Verte from Standard Form If the equation of a parabola is in the form a( h) k, then its verte is (h, k). If we complete the square on the standard equation of the parabola a b c to get it into the form a( h) k, we will find the verte in terms of a, b, and c: a b c a b a c a b b b a a a c b a b a a b b b a a c Move a a b a a b b b a a c a a a b a a ac b a Factor a out of the first two terms. and b b a a outside the parentheses. Build up the denominator. Simplif. The equation is now in the form a( h) k, and the verte is b, ac b a a. We summarize these results as follows. Parabolas in the Form a b c The graph of a b c (for a 0) is a parabola opening upward if a 0 and downward if a 0. The coordinate of the verte is b. a b You can use ac to get the coordinate of the verte, but it is usuall a easier to get it from a b c for b. We use the second approach in a Eample.
5 8 () Chapter Nonlinear Sstems and the Conic Sections E X A M P L E A calculator graph can be used to check the verte and opening of a parabola. Ais of smmetr calculator closeup = b a 0 Finding the features of a parabola from standard form Find the verte, focus, and directri of the parabola 9, and determine whether the parabola opens upward or downward. The coordinate of the verte is b 9 9 a ( ) 6. To find the coordinate of the verte, let in 9 : The verte is, 7. Because a, the parabola opens downward. To find the focus, use to get p p. The focus is of a unit below the verte at, 7 or,. The directri is the horizontal line of a unit above the verte, 7 or. 6 Smmetr and Intercepts The graph of shown in Fig..9 is said to be smmetric about the ais because the two halves of the parabola would coincide if the paper were folded on the ais. In general, the vertical line through the verte is called the ais of smmetr for the parabola. Because the coordinate of the verte for a b c is b, the ais of smmetr is b. See Fig... a a The parabola a b c has eactl one intercept. If 0 in the equation, then a(0) b(0) c c. So the intercept is (0, c). To find the intercepts, we let 0 in the equation a b c and solve the quadratic equation a b c 0. The number of intercepts ma be 0,, or, depending on the number of solutions to the quadratic equation. See Fig... FIGURE. No intercepts One intercept FIGURE. Two intercepts Graphing a Parabola When graphing a parabola, we can use the features that we have discussed to improve accurac and understanding. These features are summarized in the following bo as a strateg for graphing parabolas.
6 . The Parabola () 8 Graphing the Parabola a b c To graph the parabola a b c, use the following facts:. The parabola opens upward if a 0 and opens downward if a 0.. The coordinate of the verte is b. a. The graph is smmetric about the vertical line b. a. The intercept is (0, c).. The intercepts are found b solving a b c 0. CAUTION The focus and directri are important features of a parabola, but the are not part of the curve itself. So we do not usuall find the focus and directri when graphing a parabola from an equation. E X A M P L E helpful hint When drawing a parabola such as the one in Fig.. b hand, use our hand like a compass and draw it in two steps. If ou are righthanded, sketch the left half of the parabola first. Then turn our paper upside down to sketch the right half. = + 6 FIGURE. Graphing a parabola Determine whether the parabola 6 opens upward or downward, and find the verte, ais of smmetr, intercepts, and intercept. Find several additional points on the parabola, and sketch the graph. Because a, this parabola opens downward. The coordinate of the verte is b ( ) a ( ). If, then 6 6. So the verte is,. The ais of smmetr is the vertical line. To find the intercepts, let 0 in the original equation: ( )( ) 0 0 or 0 or The intercepts are (, 0) and (, 0). The intercept is (0, 6). Using all of this information and the additional points (, ), (, ), and (, 6), we get the graph shown in Fig... Maimum or Minimum Value of On a parabola that opens upward, the minimum value that attains is the coordinate of the verte. On a parabola that opens downward, the maimum value of is again the coordinate of the verte. We use the fact that the coordinate of the verte is b to obtain the maimum or minimum value of. a
7 86 (6) Chapter Nonlinear Sstems and the Conic Sections E X A M P L E 6 calculator closeup The graph of = ( ) supports the conclusion that the maimum value of occurs when =. Maimizing a product Find two numbers that have the maimum product, subject to the condition that their sum is. Because their sum is, we can let represent one number and represent the other. If represents their product, we can write the equation ( ) or. This equation is the equation of a parabola that opens downward. Its highest point, which is the maimum value of, occurs when b a ( ). If, then. The two numbers are and. Among the numbers with a sum of, the give the maimum product. So no two numbers with a sum of can have a product larger than. CAUTION Be sure to answer the question that is asked in a maimumminimum problem. If a b c, then the maimum (or minimum) value of is the coordinate of the verte. The value of that causes to reach its maimum (or minimum) value is the coordinate of the verte. E X A M P L E 7 Maimizing revenue A manufacturer of inline skates uses the formula R 60 to predict the weekl revenue in dollars that will be produced when the skates are priced at dollars per pair. Find the price that will produce the maimum revenue, and find the maimum possible revenue. The graph of R 60 is a parabola that opens downward as shown in Fig.. on the net page. To find the verte of the parabola, use a and b 60 in b : a () Now use 90 in R 60 to find R: R 60(90) 90 6,00 So if the skates are priced at $90 per pair, then the manufacturer will receive the maimum weekl revenue $6,00.
8 . The Parabola (7) 87 Revenue (in thousands of dollars) R Price (in dollars) FIGURE. WARMUPS True or false? Eplain our answer.. There is a parabola with focus (, ), directri, and verte (0, 0). False. The focus for the parabola is (0, ). True. The graph of ( ) is a parabola with verte (, ). True. The graph of 6 is a parabola. False. The graph of 9 is a parabola opening upward. False 6. For the verte and intercept are the same point. True 7. A parabola with verte (, ) and focus (, ) has no intercepts. True 8. The parabola 9 has no intercept. False 9. If (, ) satisfies a( ) k, then so does (, ). True 0. The parabola ( ) has onl one intercept. True. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the definition of a parabola given in this section? A parabola is the set of all points in a plane that are equidistant from a given line and a fied point not on the line.. What is the location of the verte? The verte is the midpoint of the line segment joining the focus and directri, perpendicular to the directri.. What are the two forms of the equation of a parabola? A parabola can be written in the forms a b c or a( h) k.. What is the distance from the focus to the verte in an parabola of the form a b c? The distance from the focus to the directri is p, where a (p).. How do we convert an equation of the form a b c into the form a( h) k? We use completing the square to convert a b c into a( h) k. 6. How do we convert an equation of the form a( h) k into the form a b c? To convert a( h) k into the form a b c, square the binomial, multipl b a, then add like terms. Find the verte, focus, and directri for each parabola. See Eample. 7. Verte (0, 0), focus 0, 8, directri Verte (0, 0), focus 0,, directri.
9 88 (8) Chapter Nonlinear Sstems and the Conic Sections 9. Verte (0, 0), focus (0, ), directri 0. Verte (0, 0), focus (0, ), directri. ( ) Verte (, ), focus (,.), directri.. ( ) Verte (, ), focus (, ), directri 6. ( ) 6 Verte (, 6), focus (,.7), directri 6.. ( ) Verte (, ), focus,, directri Find the equation of the parabola with the given focus and directri. See Eample.. Focus (0, ), directri 8 6. Focus (0, ), directri 7. Focus 0,, directri 8. Focus 0, 8, directri 8 9. Focus (, ), directri 6 0. Focus (, ), directri. Focus (, ), directri 8 8. Focus (, ), directri 8. Focus (,.), directri Focus, 7 8, directri 0 8 Write each equation in the form a( h) k. Identif the verte, focus, and directri of each parabola. See Eample.. 6 ( ) 8, verte (, 8), focus (, 7.7), directri ( ), verte (, ), focus (, 0.7), directri. 7. ( ), verte (, ), focus (,.87), directri ( ) 0, verte (, 0), focus, 9, directri ( ), verte (, ), focus, 7 8, directri ( ) 0, verte (, 0), focus, 9, directri 0. 0 ( ) 80, verte (, 80), focus, , directri , verte,, focus, 9 9 8, directri 0 8 Find the verte, focus, and directri of each parabola (without completing the square), and determine whether the parabola opens upward or downward. See Eample.. Verte (, ), focus,, directri, upward. 6 7 Verte (, 6), focus,, directri 6, upward. Verte (, ), focus,, directri, downward 6. 9 Verte (, ), focus,, directri, downward 7. 6 Verte (, ), focus,, directri, upward 8. Verte (, ), focus, 7 8, directri 8, upward 9. Verte, 7 downward 0., focus,, directri 9, Verte,, focus,, directri, downward. Verte (0, ), focus 0,, directri, upward. 6 Verte (0, 6), focus 0, 6 8, directri 7 8, downward
10 . The Parabola (9) 89 Find the verte, ais of smmetr, intercepts, and intercept for each parabola. Find several additional points on the parabola, and then sketch its graph. See Eample ais of Verte (, ), ais of Verte, smmetr smmetr, intercepts (0, ), (, 0), (, 0), intercepts (0, 8), (, 0), (, 0) Verte (, 0), ais of Verte (, 0), ais of smmetr, smmetr, intercepts (0, ), (, 0) intercepts (0, 9), (, 0).. 9 Verte, 0, ais of Verte, 0, ais of Verte (, 9), ais of Verte (, 6), ais of smmetr, smmetr, intercepts (0, 8), (, 0), intercepts (0, ), (, 0) (, 0), (, 0) smmetr, intercepts (0, ),, 0 smmetr, intercepts, 0, (0, 9) 7. ( ) 8. ( ) Verte (, ), ais of Verte (, ), ais of smmetr, smmetr, intercept (0, ) intercepts (0, ), 6, 0, 6, 0. Verte is,, ais of smmetr, intercepts (0, 0), (, 0). 9 Verte, 7, ais of smmetr, intercepts (0, 0), (, 0)
11 90 (0) Chapter Nonlinear Sstems and the Conic Sections. 6. Verte (0, ), ais of Verte (0, ), ais of smmetr 0, smmetr 0, intercept (0, ) intercepts (0, ), 6, 0, 6, Verte (, ), ais of Verte (, ), ais of smmetr, smmetr, interintercepts (0, ), cepts (0, ), (, 0), (, 0), (, 0) (, 0) 6. Maimum product. What is the maimum product that can be obtained b two numbers that a have a sum of 6? 9 6. Maimum area. A gardener plans to enclose a rectangular area with 60 feet (ft) of fencing. What dimensions for the length and width would maimize the area? 0 ft b 0 ft 6. Maimum area. Another gardener has a footwide gate for her rectangular garden in place. If she has 00 ft of fencing in addition to the gate, then what dimensions for the length and width would maimize the area? 6 ft b 6 ft 6. Minimum hpotenuse. If the total length of the legs of a right triangle is 6 ft, then what lengths for the legs would minimize the square of the hpotenuse? ft each 66. Maimum revenue. A concert promoter uses the formula R 700p 0p to calculate the total revenue for a concert if the ticket price is p dollars each. What ticket price will maimize the revenue? See the accompaning figure. $ 9. ( ) 60. ( ) Verte (, 0), ais of Verte (, ), ais of smmetr, smmetr, intercepts (0, ), (, 0) intercepts (0, ),, 0,, 0 Revenue from concert tickets (in thousands of dollars) R Price of tickets (in dollars) FIGURE FOR EXERCISE Maimum height. If a punter kicks a football straight up at a velocit of 8 feet per second (ft/sec) from a height of 6 ft, then the football s distance above the earth after t seconds is given b the formula s 6t 8t 6. What is the maimum height reached b the ball? How long does the ball take to reach its maimum height? Could the punter Solve each problem. See Eamples 6 and Maimum product. Find two numbers that have the maimum possible product among numbers that have a sum of 8. and Height of football (ft) s t Time (sec) FIGURE FOR EXERCISE 67
12 . The Parabola () 9 hit the roof of the New Orleans Superdome, which is 7 ft above the plaing field? 6 ft, sec, no 68. Maimum height. If a baseball is hit straight upward at 0 ft/sec from a height of ft, then its distance above the earth after t seconds is given b the formula s 6t 0t. What is the maimum height attained b the ball? How long does the ball take to reach its maimum height? 6.6 ft,.687 sec v 0 ft/sec 7. Arecibo Observator. The largest radio telescope in the world uses a,000ft parabolic dish, suspended in a valle in Arecibo, Puerto Rico. The antenna hangs above the verte of the dish on cables stretching from two towers. The accompaning figure shows a cross section of the parabolic dish and the towers. Assuming the verte is at (0, 0), find the equation for the parabola. Find the distance from the verte to the antenna located at the focus ,. ft Antenna at focus ft 00 ft 00 ft FIGURE FOR EXERCISE Minimum cost. It costs Acme Manufacturing C dollars per hour to operate its golf ball division. An analst has determined that C is related to the number of golf balls produced per hour,, b the equation C What number of balls per hour shouldacme produce to minimize the cost per hour of manufacturing these golf balls? Maimum profit. A chain store manager has been told b the main office that dail profit, P, is related to the number of clerks working that da,, according to the equation P 00. What number of clerks will maimize the profit, and what is the maimum possible profit? 6 clerks, $ World s largest telescope. The largest reflecting telescope in the world is the 6meter (m) reflector on Mount Pastukhov in Russia. The accompaning figure shows a cross section of a parabolic mirror 6 m in diameter with the verte at the origin and the focus at (0, ). Find the equation of the parabola. 6 0 (0, ) 000 ft FIGURE FOR EXERCISE 7 Graph both equations of each sstem on the same coordinate aes. Use elimination of variables to find all points of intersection (, ), (, ) (, ), (, ) (, ) (, 6) 6 m FIGURE FOR EXERCISE 7
13 9 () Chapter Nonlinear Sstems and the Conic Sections ,, (, 0),, (, 0) 8. (0, 0) (, ) (, 0) 8. (, 0) (0, ) (, ) (, ), (, 6) (, 6), (, ) Write an equation in the form a b c for each given parabola. 8. (0, ) (, 0) (, 0) 8. (, ) (, ) (0, 0) GETTING MORE INVOLVED 8. Eploration. Consider the parabola with focus ( p, 0) and directri p for p 0. Let (, ) be an arbitrar point on the parabola. Write an equation epressing the fact that the distance from (, ) to the focus is equal to the distance from (, ) to the directri. Rewrite the equation in the form a, where a. p 86. Eploration. In general, the graph of a( h) k for a 0 is a parabola opening left or right with verte at (k, h). a) For which values of a does the parabola open to the right, and for which values of a does it open to the left? b) What is the equation of its ais of smmetr? c) Sketch the graphs ( ) and ( ). a) Right for a 0 and left for a 0 b) h GRAPHING CALCULATOR EXERCISES 87. Graph using the viewing window with and 0. Net graph using the viewing window and 7. Eplain what ou see. The graphs have identical shapes. 88. Graph and 6 9 in the viewing window and 0. Does the line appear to be tangent to the parabola? Solve the sstem and 6 9 to find all points of intersection for the parabola and the line. Intersection (, 9)
THE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More information10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it
3330_00.qd /8/05 9:00 AM Page 735 Section 0. Introduction to Conics: Parabolas 735 0. Introduction to Conics: Parabolas What ou should learn Recognize a conic as the intersection of a plane and a doublenapped
More informationName Class Date. Deriving the Equation of a Parabola
Name Class Date Parabolas Going Deeper Essential question: What are the defining features of a parabola? Like the circle, the ellipse, and the hperbola, the parabola can be defined in terms of distance.
More informationSECTION 91 Conic Sections; Parabola
66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships
More informationDeriving the StandardForm Equation of a Parabola
Name Class Date. Parabolas Essential Question: How is the distance formula connected with deriving equations for both vertical and horizontal parabolas? Eplore Deriving the StandardForm Equation of a
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationQuadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it
0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form
More informationThe Parabola. The Parabola in Terms of a Locus of Points
The Parabola Appolonius of Perga (5 B.C.) discovered that b intersecting a right circular cone with a plane slanted the same as the side of the cone, (formall, when it is parallel to the slant height),
More information3.1 Quadratic Functions
33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial
More informationQuadratic Functions. MathsStart. Topic 3
MathsStart (NOTE Feb 2013: This is the old version of MathsStart. New books will be created during 2013 and 2014) Topic 3 Quadratic Functions 8 = 3 2 6 8 ( 2)( 4) ( 3) 2 1 2 4 0 (3, 1) MATHS LEARNING CENTRE
More informationAlgebra 2 Honors: Quadratic Functions. Student Focus
Resources: SpringBoard Algebra Online Resources: Algebra Springboard Tet Algebra Honors: Quadratic Functions Semester 1, Unit : Activit 10 Unit Overview In this unit, students write the equations of quadratic
More informationYears 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 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationGraphing Quadratic Functions
A. THE STANDARD PARABOLA Graphing Quadratic Functions The graph of a quadratic function is called a parabola. The most basic graph is of the function =, as shown in Figure, and it is to this graph which
More information10.1. 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 informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More information10.2 THE QUADRATIC FORMULA
10. The Quadratic Formula (10 11) 535 100. Eploration. Solve k 0 for k 0,, 5, and 10. a) When does the equation have only one solution? b) For what values of k are the solutions real? c) For what values
More informationCourse 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook
Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root
More information41. Quadratic Functions and Transformations. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
41 Quadratic Functions and Transformations Vocabular Review 1. Circle the verte of each absolute value graph. Vocabular Builder parabola (noun) puh RAB uh luh Related Words: verte, ais of smmetr, quadratic
More information7.3 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 information8.7 The Parabola. PF = PD The fixed point F is called the focus. The fixed line l is called the directrix.
8.7 The Parabola The Hubble Space Telescope orbits the Earth at an altitude of approimatel 600 km. The telescope takes about ninet minutes to complete one orbit. Since it orbits above the Earth s atmosphere,
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More information4 NonLinear relationships
NUMBER AND ALGEBRA NonLinear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas
More information3.1 Quadratic Functions
Section 3.1 Quadratic Functions 1 3.1 Quadratic Functions Functions Let s quickl review again the definition of a function. Definition 1 A relation is a function if and onl if each object in its domain
More informationAlgebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Algebra Module A7 The Parabola Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED December, The Parabola Statement of Prerequisite Skills Complete
More information3 Quadratic Functions
3 Quadratic Functions 3.1 Transformations of Quadratic Functions 3. Characteristics of Quadratic Functions 3.3 Focus of a Parabola 3. Modeling with Quadratic Functions SEE the Big Idea Meteorologist (p.
More informationSection C Non Linear Graphs
1 of 8 Section C Non Linear Graphs Graphic Calculators will be useful for this topic of 8 Cop into our notes Some words to learn Plot a graph: Draw graph b plotting points Sketch/Draw a graph: Do not plot,
More informationSection 72 Ellipse. Definition of an Ellipse The following is a coordinatefree definition of an ellipse: DEFINITION
7 Ellipse 3. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line () 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More informationConics and Polar Coordinates
Contents 17 Conics and Polar Coordinates 17.1 Conic Sections 2 17.2 Polar Coordinates 23 17.3 Parametric Curves 33 Learning outcomes In this Workbook ou will learn about some of the most important curves
More informationREVIEW 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 informationAttributes and Transformations of Quadratic Functions VOCABULARY. Maximum value the greatest yvalue of a function. Minimum value the least
 Attributes and Transformations of Quadratic Functions TEKS FCUS TEKS ()(B) Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction of opening.
More informationD.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 information17.1. Conic Sections. Introduction. Prerequisites. Learning Outcomes. Learning Style
Conic Sections 17.1 Introduction The conic sections (or conics)  the ellipse, the parabola and the hperbola  pla an important role both in mathematics and in the application of mathematics to engineering.
More informationFinding Complex Solutions of Quadratic Equations
COMMON CORE y  0 y   0  Locker LESSON 3.3 Finding Comple Solutions of Quadratic Equations Name Class Date 3.3 Finding Comple Solutions of Quadratic Equations Essential Question: How can you find the
More informationSection 23 Quadratic Functions
118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the
More informationCollege Algebra  MAT 161 Page: 1 Copyright 2009 Killoran
College Algera  MAT 161 Page: 1 Copright 009 Killoran Quadratic Functions The graph of f./ D a C C c (where a,,c are real and a 6D 0) is called a paraola. Paraola s are Smmetric over the line that passes
More information6.3 Parametric Equations and Motion
SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh
More informationIf (a)(b) 5 0, then a 5 0 or b 5 0.
chapter Algebra Ke words substitution discriminant completing the square real and distinct imaginar rational verte parabola maimum minimum surd irrational rationalising the denominator Section. Quadratic
More informationTranslating Points. Subtract 2 from the ycoordinates
CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that
More information25. The Graph of y = kx 2. Vocabulary. Rates of Change. Lesson. Mental Math
Chapter 2 Lesson 25 The Graph of = k 2 BIG IDEA The graph of the set of points (, ) satisfing = k 2, with k constant, is a parabola with verte at the origin and containing the point (1, k). Vocabular
More informationD.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 informationSection 105 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 informationHOW TO COMPLETE THE SQUARE
HOW TO COMPLETE THE SQUARE LANCE D. DRAGER. Introduction To complete the square means to take a quadratic function f() = a 2 + b + c and do the algebra to write it in the form f() = a( h) 2 + k, for some
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More information8 Graphs of Quadratic Expressions: The Parabola
8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = 2 + 1 was a straight line. The graph of a function which is not linear therefore cannot be
More information4Unit 2 Quadratic, Polynomial, and Radical Functions
CHAPTER 4Unit 2 Quadratic, Polnomial, and Radical Functions Comple Numbers, p. 28 f(z) 5 z 2 c Quadratic Functions and Factoring Prerequisite Skills... 234 4. Graph Quadratic Functions in Standard Form...
More informationBerkeley City College Precalculus w/ Analytic Geometry  Math 2  Chapters 13 Pt 2 Homework 2 Due: Name
Berkele Cit College Precalculus w/ Analtic Geometr  Math 2  Chapters 13 Pt 2 Homework 2 Due: Name Graph the function. 1) f() = + 1 if < 14 if 1 1)   Objective: (2.4) Graph Piecewisedefined Functions
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationLocus and the Parabola
Locus and the Parabola TERMINOLOGY Ais: A line around which a curve is reflected eg the ais of symmetry of a parabola Cartesian equation: An equation involving two variables and y Chord: An interval joining
More information2.3 Quadratic Functions
. Quadratic Functions 9. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions: the
More informationAdditional Topics in Analytic Geometry
bar1969_ch11_961984.qd 17/1/08 11:43 PM Page 961 Pinnacle ju111:venus:mhia06:mhia06:student EDITION:CH 11: CHAPTER Additional Topics in Analtic Geometr C ANALYTIC geometr is the stud of geometric objects
More informationSECTION 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 information2 Analysis of Graphs of
ch.pgs116 1/3/1 1:4 AM Page 1 Analsis of Graphs of Functions A FIGURE HAS rotational smmetr around an ais I if it coincides with itself b all rotations about I. Because of their complete rotational smmetr,
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationFamilies of Quadratics
Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish
More informationInvestigation. So far you have worked with quadratic equations in vertex form and general. Getting to the Root of the Matter. LESSON 9.
DA2SE_73_09.qd 0/8/0 :3 Page Factored Form LESSON 9.4 So far you have worked with quadratic equations in verte form and general form. This lesson will introduce you to another form of quadratic equation,
More informationThe Quadratic Function
0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral
More informationArchimedes quadrature of the parabola and the method of exhaustion
JOHN OTT COLLEGE rchimedes quadrature of the parabola and the method of ehaustion CLCULUS II SCIENCE) Carefull stud the tet below and attempt the eercises at the end. You will be evaluated on this material
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationAnswers (Anticipation Guide and Lesson 101)
Answers (Anticipation Guide and Lesson 0) Lesson 0 Copright Glencoe/McGrawHill, a division of The McGrawHill Companies, Inc. 0 NAME DATE PERID Lesson Reading Guide Midpoint and Distance Formulas Get
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationApplications of Quadratic Functions Word Problems ISU. Part A: Revenue and Numeric Problems
Mr. Gardner s MPMD Applications of Quadratic Functions Word Problems ISU Part A: Revenue and Numeric Problems When ou solve problems using equations, our solution must have four components: 1. A let statement,
More information2.3 Quadratic Functions
88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:
More informationCoordinate Geometry. Positive gradients: Negative gradients:
8 Coordinate Geometr Negative gradients: m < 0 Positive gradients: m > 0 Chapter Contents 8:0 The distance between two points 8:0 The midpoint of an interval 8:0 The gradient of a line 8:0 Graphing straight
More informationIn this section you will learn how to draw the graph of the quadratic function defined by the equation. f(x) = a(x h) 2 + k. (1)
Section.1 The Parabola 419.1 The Parabola In this section ou will learn how to draw the graph of the quadratic function defined b the equation f() = a( h) 2 + k. (1) You will quickl learn that the graph
More informationUnit 1 Quadratic Functions & Equations
1 Unit 1 Quadratic Functions & Equations Graphing Quadratics Part I: What is a Quadratic? A quadratic is an epression of degree. E) Graph of y : The most basic quadratic function is given by y. Create
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0 P(X, Y) X
Rotation of Aes For a discussion of conic sections, see Appendi. In precalculus or calculus ou ma have studied conic sections with equations of the form A C D E F Here we show that the general seconddegree
More informationSECTION 25 Combining Functions
2 Combining Functions 16 91. Phsics. A stunt driver is planning to jump a motorccle from one ramp to another as illustrated in the figure. The ramps are 10 feet high, and the distance between the ramps
More informationTHE POINTSLOPE FORM
. The PointSlope Form () 67. THE POINTSLOPE FORM In this section In Section. we wrote the equation of a line given its slope and intercept. In this section ou will learn to write the equation of a
More informationQuadratic Functions. Academic Vocabulary justify derive verify
Quadratic Functions 015 College Board. All rights reserved. Unit Overview This unit focuses on quadratic functions and equations. You will write the equations of quadratic functions to model situations.
More informationQuadratic Functions and Parabolas
MATH 11 Quadratic Functions and Parabolas A quadratic function has the form Dr. Neal, Fall 2008 f () = a 2 + b + c where a 0. The graph of the function is a parabola that opens upward if a > 0, and opens
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More information17.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,
More informationconics and polar coordinates
17 Contents nts conics and polar coordinates 1. Conic sections 2. Polar coordinates 3. Parametric curves Learning outcomes In this workbook ou will learn about some of the most important curves in the
More informationPolynomial and Rational Functions
Chapter Section.1 Quadratic Functions Polnomial and Rational Functions Objective: In this lesson ou learned how to sketch and analze graphs of quadratic functions. Course Number Instructor Date Important
More informationPolynomial 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 informationChapter 7: Eigenvalues and Eigenvectors 16
Chapter 7: Eigenvalues and Eigenvectors 6 SECION G Sketching Conics B the end of this section ou will be able to recognise equations of different tpes of conics complete the square and use this to sketch
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationLines 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 information11.7 MATHEMATICAL MODELING WITH QUADRATIC FUNCTIONS. Objectives. Maximum Minimum Problems
a b Objectives Solve maimum minimum problems involving quadratic functions. Fit a quadratic function to a set of data to form a mathematical model, and solve related applied problems. 11.7 MATHEMATICAL
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a
More informationFilling in Coordinate Grid Planes
Filling in Coordinate Grid Planes A coordinate grid is a sstem that can be used to write an address for an point within the grid. The grid is formed b two number lines called and that intersect at the
More informationTransformations of Function Graphs
   0        Locker LESSON.3 Transformations of Function Graphs Teas Math Standards The student is epected to: A..C Analze the effect on the graphs of f () = when f () is replaced b af (), f (b),
More informationGRAPHS OF RATIONAL FUNCTIONS
0 (0) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique
More informationGraph each function. Compare to the parent graph. State the domain and range. 1. SOLUTION:
 Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
More informationFACTORING ax 2 bx c WITH a 1
296 (6 20) Chapter 6 Factoring 6.4 FACTORING a 2 b c WITH a 1 In this section The ac Method Trial and Error Factoring Completely In Section 6.3 we factored trinomials with a leading coefficient of 1. In
More information1 Quadratic Functions
C h a p t e r 1 Quadratic Functions Quadratic Functions Eplain the meaning of the term function, and distinguish a function from a relation that is not a function, through investigation of linear and quadratic
More informationQUADRATIC FUNCTIONS AND COMPLEX NUMBERS
CHAPTER 86 5 CHAPTER TABLE F CNTENTS 5 Real Roots of a Quadratic Equation 52 The Quadratic Formula 53 The Discriminant 54 The Comple Numbers 55 perations with Comple Numbers 56 Comple Roots of a
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationEssential Question How can you use completing the square to solve a quadratic equation?
9.4 Solving Quadratic Equations Completing the Square Essential Question How can ou use completing the square to solve a quadratic equation? Work with a partner. a. Write the equation modeled the algera
More information