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. A parabola is the set of all points in a plane that are the same distance from a fied point, called the focus, and a fied line, called the directri. The midpoint of the shortest segment connecting the focus and the directri is the verte of the parabola. The ais of smmetr is a line perpendicular to the directri and passes through the focus and the verte. 12-5 1 G-GPE.1.2 EXPLORE Deriving the Equation of a Parabola Focus Houghton Mifflin Harcourt Publishing Compan Use distance on the coordinate plane to find the equation of a parabola. A In the figure, P(, ) on the parabola is equidistant from the focus F (0, p) and the directri = -p. A line perpendicular to the directri from P intersects the directri at D(, -p). ( - 1 ) 2 + ( - 1 ) 2 = ( - 2 ) 2 + ( - 2 ) 2 Distance Formula 2 + 2 = 2 2 + + 2 = 2 Simplif. 2 2 + = Substitute (0, p) for ( 1, 1 ) and (, -p) for ( 2, 2 ). Square both sides. + - + = + + Epand the binomials. - = Subtract 2 and p 2 from both sides. 2 = = ( 2 ) Solve for. Add 2p to both sides. This is the standard form of the equation of a parabola. Sometimes, the equation is given in the form 2 =. Verte = -p F(0, p) P(, ) Ais of Smmetr Directri B A parabola has its focus at F (0, 6) and directri = -6. The equation of the parabola is =. C The focus of the parabola 2 = 18 is. The directri is. Chapter 12 687 Lesson 5
REFLECT 1. In the first step of the derivation of the equation of a parabola, the points (0, p) and (, -p) were substituted for ( 1, 1 ) and ( 2, 2 ), ielding the equation ( - 0) 2 + ( - p) 2 = ( - ) 2 + ( + p) 2. Compare that equation with the equation ou would write if the parabola were rotated 90 so that its ais of smmetr were horizontal, as in the figure at the right. How would the remaining equations in the derivation be affected? What would the derivation ield as the standard form of the equation of a parabola with a horizontal ais of smmetr? Verte Ais of Smmetr = -p Directri F(p, 0) P(, ) Focus The standard form of the equation of a parabola with a vertical ais is = 1 4p 2. The standard form of the equation of a parabola with a horizontal ais is = 1 4p 2. 2 G-GPE.1.2 EXAMPLE Finding the Equation of a Parabola with its Verte at the Origin Write the equation of each parabola in standard form. A B = 0.5 REFLECT F(-3, 0) = 3 F(0, -0.5) Houghton Mifflin Harcourt Publishing Compan 2. Describe how the value of p relates to the shape of the parabola. Chapter 12 688 Lesson 5
For the circle, the ellipse, and the hperbola, translating the center of the conic section from (0, 0) to (h, k) changed in the standard form of the equation of the figure to - h and changed to - k. A similar transformation takes place when the verte of a parabola is translated from (0, 0) to (h, k). Direction of Ais of Smmetr Equation of Ais of Smmetr Standard Form of the Equation of a Parabola with Verte at (h, k) Vertical = h - k = 1 4p ( - h) 2 Horizontal = k - h = 1 4p ( - k) 2 3 G-GPE.1.2 EXAMPLE Finding the Equation of a Parabola with its Verte Not at the Origin Use the focus and directri to sketch the parabola. Then find the equation of the parabola. A A parabola has focus (3, 8) and directri = 4. The verte of the parabola is. So h = and k =. Use the fact that p equals the distance from the focus to the verte to find p: p =. Standard form of the equation of the parabola: Houghton Mifflin Harcourt Publishing Compan B A parabola has focus (3, -1) and directri = -2 The verte of the parabola is. So h = and k =. Find p: p =. Standard form of the equation of the parabola: REFLECT 3. During the stud of quadratic functions, the standard form of a quadratic function is given as = a 2 + b + c. Find values of a, b, and c which show that this equation is equivalent to the standard form given in this lesson - k = 1 4p ( - h) 2. Eplain how ou found the values. Chapter 12 689 Lesson 5
PRACTICE Write the equation of each parabola in standard form. Focus Directri Equation 1. F (2, 0) = -2 2. F (0, 8) = -8 3. F (-20, 0) = 20 4. F ( 0, - 1 12) = 1 12 5. F (5, 5) = -3 6. F (3, 0) = -2 7. F (4, -3) = 6 8. F (8, 0) = 4 9. F (10, -3) = 5 10. F (6, 2) = 4 Houghton Mifflin Harcourt Publishing Compan 11. F (7, -7) = -2 12. F (-1, 2) = -1 Chapter 12 690 Lesson 5
12-5 Name Class Date Date Name Class Practice Additional Practice 12-5 Parabolas LESSON Use the Distance Formula to find the equation of a parabola with the given focus and directri. 1. F(6, 0), = 3 2. F(1, 0), = 4 _ Write the equation in standard form for each parabola. 3. Verte (0, 0), directri = 2 4. Verte (0, 0), focus (9, 0) _ 5. Focus ( 6, 0), directri = 6 6. Verte (0, 0), focus (0, 3) _ Find the verte, value of p, ais of smmetr, focus, and directri of each parabola. Then graph. 7. 1 = 1 2 12 8. + 2 = Houghton Mifflin Harcourt Publishing Compan _ 1 ( 1)2 4 Solve. 9. A spotlight has parabolic cross sections. a. Write an equation for a cross section of the spotlight if the bulb is 6 inches from the verte and the verte is placed at the origin. b. If the spotlight has a diameter of 36 inches at its opening, find the depth of the spotlight if the bulb is 6 inches from the verte. Chapter 12 Copright b Holt McDougal. Additions and changes691 Lesson 5 Original content to the original content are the responsibilit of the instructor. 80 Holt McDougal Algebra 2
Problem Solving += += + += + = ( ) + = ( ) + = ( ) + = Houghton Mifflin Harcourt Publishing Compan Chapter 12 692 Lesson 5