Quiz #20. y 2 + 6y = 12x y 2 + 6y + 9 = 12x (y + 3) 2 = 12x + 24 (y + 3) 2 = 12(x 2)

Transcription

1 Math 25 Trigonometr Ma 2, 2017 Qiz #20 Name: Answer Ke Daid Arnold Instrctions. (10 points) This qiz is open notes, open book. This incldes an spplementar tets or online docments. Yo mst answer all of the eercises on or own. Yo are not allowed to work in grops or pairs on the qiz. Yo are not allowed to enlist the aid of a ttor, classmate, or friend to help with the qiz. All work mst be or own. Honor Pledge: I promise that the soltion sbmitted was done entirel b me. I receied no help from colleages, friends, or ttors. I also did not share an parts of m soltion with an of m classmates. Signatre: Please staple this qiz and honor pledge atop or soltions. (5 pts ) 1. Consider the eqation = 0. Complete the sqare to pt the eqation into the appropriate form, then plot the reslt on graph paper. Inclde the directri and label it with its eqation. Inclde the focs and erte and label them with their coordinates. Draw the lats rectm. Label and scale each ais. All lines mst be drawn with a rler. Soltion: Start b completing the sqare. We anticipate the form 2 = 4a, so moe all terms containing to the left-hand side of the eqation and all other terms to the right-hand side = 0 Half of 6 is 3, sqare to get 9. Add 9 to both sides = = ( + 3) 2 = ( + 3) 2 = 12( 2) Comparing 2 = 4a with 2 = 12 reeals that 4a = 12, so a = 3. Howeer, becase the eqation ( + 3) 2 = 12( 2) replaces with 2 and with + 3, the graph will shift two nits to the right and three nits downward. We ll begin b shifting the coordinate aes, then making all of or measrements from the new coordinate aes.

2 Math 25 Trigonometr/Qiz #20 Page 2 of 6 Name: Answer Ke Becase or eqation has the form 2 = 4a, with a = 3, it opens to the left. Ths, the focs is at F ( 1, 3) and the directri is the line = 5. F ( 1, 3) = 5 The length of the lats rectm is 4a = 12 = 12, so we mst trael 6 nits p and down from the focs.

3 Math 25 Trigonometr/Qiz #20 Page 3 of 6 Name: Answer Ke F ( 1, 3) = 5 We can now draw the parabola F ( 1, 3) 11 = 5 (5 pts ) 2. Consider the eqation = 0. Complete the sqare to pt the eqation into the appropriate form, then plot the reslt on graph paper. Inclde the foci and intercepts and label them with their eact intercepts, althogh o shold approimate the foci coordinates so that o get them in approimatel the correct place on or graph. All lines mst be drawn with a rler.

4 Math 25 Trigonometr/Qiz #20 Page 4 of 6 Name: Answer Ke Soltion: Start b completing the sqare = 0 9( ) + 4( 2 2+ ) = 4 9( ) + 4( ) = ( + 2) 2 + 4( 1) 2 = 36 ( + 2) ( 1)2 9 Becase is replaced with + 2 and is replaced b 1, the ellipse is shifted 2 nits to the left and 1 nit pward, ptting the center of the ellipse at ( 2, 1). We will draw a new set of coordinate aes in red, which represents the shifted coordinate sstem. These will help s make the correct measrements. = 1 ( 2, 1) Note that in ( + 2) 2 /4 + ( 1) 2 /9 = 1, the larger nmber a 2 = 9 is nder the term. Ths, we go a = ±3 nits in the ertical direction from the center of the ellipse. Becase b 2 = 4, we go b = ±2 nits in the horizontal direction from the center of the ellipse.

5 Math 25 Trigonometr/Qiz #20 Page 5 of 6 Name: Answer Ke ( 2, 4) ( 4, 1) ( 2, 1) (0, 1) ( 2, 2) To find the foci, we se: b 2 = a 2 c 2 4 = 9 c 2 c 2 = 5 c = ± 5 The foci lie on the major ais, in this case the ertical ais Ths we mst walk c = ± 5 nits pward and downward from the center of the ellipse to locate the foci, which pts s approimatel at ( 2, 3.23) and ( 2, 1.23), bt we label these foci with their eact coordinates.

6 Math 25 Trigonometr/Qiz #20 Page 6 of 6 Name: Answer Ke 5 ( 2, 4) ( 2, 1 + 5) ( 4, 1) ( 2, 1) (0, 1) 2 ( 2, 1 5) ( 2, 2)