Write equations of parabolas in standard form. Graph parabolas. are parabolas used in manufacturing?

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1 Parabolas Vocabular parabola conic section focus directri latus rectum Write equations of parabolas in standard form. Graph parabolas. are parabolas used in manufacturing? A mirror or other reflective object in the shape of a parabola has the propert that parallel incoming ras are all reflected to the same point. r, if that point is the source of ras, the ras become parallel when the are reflected. EQUATINS F PARABLAS In Chapter 6, ou learned that the graph of an equation of the form a 2 b c is a parabola. A parabola can also be obtained b slicing a double cone on a slant as shown below on the left. An figure that can be obtained b slicing a double cone is called a conic section. ther conic sections are also shown below. parabola circle ellipse hperbola Stud Tip Focus of a Parabola The focus is the special point referred to at the beginning of the lesson. A parabola can also be defined as the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directri. The parabola at the right has its focus at (2, 3), and the equation of its directri is. You can use the Distance Formula to find an equation of this parabola. focus (2, 3) (, ) directri (, ) Let (, ) be an point on this parabola. The distance from this point to the focus must be the same as the distance from this point to the directri. The distance from a point to a line is measured along the perpendicular from the point to the line. distance from (, ) to (2, 3) distance from (, ) to (, ) ( ) 2 2 ( 3) 2 ( ) 2 [)] ( 2 ( 2) 2 ( 3) ( ) 2 Square each side. ( 2) Square 3 and. ( 2) Isolate the -terms. 8 ( 2)2 Divide each side b 8. Lesson 8-2 Parabolas 49

2 An equation of the parabola with focus at (2, 3) and directri with equation is 8 ( 2)2. The equation of the ais of smmetr for this parabola is 2. The ais of smmetr intersects the parabola at a point called the verte. The verte is the point where the graph turns. The verte of this parabola is at (2, ). Since 8 is positive, the parabola opens upward. An equation of the form a 2 b c can be written in standard form. The standard form of the equation of a parabola with verte (h, k) and ais of smmetr h is a( h) 2 k. If a 0, k is the minimum value of the related function and the parabola opens upward. If a 0, k is the maimum value of the related function and the parabola opens downward. Equation of a Parabola verte h ais of smmetr (h, k) Eample Analze the Equation of a Parabola Write in standard form. Identif the verte, ais of smmetr, and direction of opening of the parabola. Stud Tip Look Back To review completing the square, see Lesson ( 2 8) 50 3( 2 8 ) 50 3( ) 3( 2 8 6) 50 3(6) 3( 4) 2 2 riginal equation Factor 3 from the -terms. Complete the square on the right side. The 6 added when ou complete the square is multiplied b 3. 3[ (4)] 2 2 (h, k) (4, 2) The verte of this parabola is located at (4, 2), and the equation of the ais of smmetr is 4. The parabola opens upward. GRAPH PARABLAS You can use smmetr and translations to graph parabolas. The equation a( h) 2 k can be obtained from a 2 b replacing with h and with k. Therefore, the graph of a( h) 2 k is the graph of a 2 translated h units to the right and k units up. Notice that each side of the graph is the reflection of the other side about the -ais. Eample 2 Graph each equation. a. 2 2 Graph Parabolas For this equation, h 0 and k 0. The verte is at the origin. Since the equation of the ais of smmetr is 0, substitute some small positive integers for and find the corresponding -values. Since the graph is smmetric about the -ais, the points at (, 2), (2, 8), and (3, 8) are also on the parabola. Use all of these points to draw the graph Chapter 8 Conic Sections

3 b. 2( 2) 2 3 The equation is of the form a( h) 2 k, where h 2 and k 3. The graph of this equation is the graph of 2 2 in part a translated 2 units to the right and up 3 units. The verte is now at (2, 3) You can use paper folding to investigate the characteristics of a parabola. Parabolas TEACHING TIP Model Step Start with a sheet of wa paper that is about 5 inches long and 2 inches wide. Make a line that is perpendicular inch to the sides of the sheet b folding the sheet near one end. pen up the paper focus again. This line is the directri. Mark a point about midwa between the sides directri of the sheet so that the distance from the directri is about inch. This point 5 inches is the focus. Put the focus on top of an point on the directri and crease the paper. Make about 20 more creases b placing the focus on top of other points on the directri. The lines form the outline of a parabola. Step 2 Start with a new sheet of wa paper. Form another outline of a parabola with a focus that is about 3 inches from the directri. Step 3 n a new sheet of wa paper, form a third outline of a parabola with a focus that is about 5 inches from the directri. Analze Compare the shapes of the three parabolas. How does the distance between the focus and the directri affect the shape of a parabola? The shape of a parabola and the distance between the focus and directri depend on the value of a in the equation. The line segment through the focus of a parabola and perpendicular to the ais of smmetr is called the latus rectum. The endpoints of the latus rectum lie on the parabola. In the figure at the right, the latus rectum is AB. The length of the latus rectum of the parabola with equation a( h) 2 k is a units. The endpoints of the latus rectum are 2a units from the focus. latus rectum A focus directri f (h, k 4a ) ais of smmetr V(h, k) B Lesson 8-2 Parabolas 42

4 Equations of parabolas with vertical aes of smmetr are of the form a( h) 2 k and are functions. Equations of parabolas with horizontal aes of smmetr are of the form a( k) 2 h and are not functions. Information About Parabolas Form of Equation a( h) 2 k a( k) 2 h Verte (h, k) (h, k) Ais of Smmetr h k Focus h, k 4 a h 4 a, k Directri k 4 a h 4a Direction of pening upward if a 0, right if a 0, downward if a 0 left if a 0 Length of Latus Rectum a units a units d D Satellite TV The important characteristics of a satellite dish are the diameter D, depth d, and the ratio D f, where f is the distance between the focus and the verte. A tpical dish has the values D 60 cm, d 6.25 cm, and D f 0.6. Source: Eample 3 Graph Graph an Equation Not in Standard Form First, write the equation in the form a( k) 2 h ( 2 2 ) 3 Complete the square. There is a 2 term, so isolate the and 2 terms. 4 ( 2 2 ) 3 Add and subtract, since ( ) ( )2 3 Write 2 2 as a square. (h, k) (3, ) Then use the following information to draw the graph. verte: (3, ) ais of smmetr: focus: 3, or (4, ) 4 4 directri: 3 or direction of opening: right, since a 0 length of latus rectum: 4 Eample 4 SATELLITE TV or 4 units (3, ) 2 (4, ) Remember that ou can plot as man points as necessar to help ou draw an accurate graph. Write and Graph an Equation for a Parabola Satellite dishes have parabolic cross sections. a. Use the information at the left to write an equation that models a cross section of a satellite dish. Assume that the focus is at the origin and the parabola opens to the right. First, solve for f. Since D f 0.6, and D 60, f 0.6(60) or 36. The focus is at (0, 0), and the parabola opens to the right. So the verte must be at (36, 0). Thus, h 36 and k 0. Use the -coordinate of the focus to find a. 422 Chapter 8 Conic Sections

5 a h 36; The -coordinate of the focus is a Add 36 to each side. 44a Multipl each side b 4a. a Divide each side b An equation of the parabola is b. Graph the equation. The length of the latus rectum is or 44 units, so 44 the graph must pass through (0, 72) and (0, 72). According to the diameter and depth of the dish, the graph must pass through (29.75, 30) and (29.75, 30). Use these points and the information from part a to draw the graph