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# The Mr. K question

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### The Mr. K question

1. 1. Picture by flickr user Ahmed Rabea
2. 2. Our Nefarious Plan • In order to distract Mr. K from his coffee we threw his block of wood into the hallway. When he went to pick it up he discovered a smart board with two tricky mathematical questions on it. Upon looking at the solution Mr. K realized there were a few errors! Being the math teacher that he is, Mr. K could not let those errors slide so he decided to fix the answer, as we knew he would, giving us time to steal his coffee!! • NOTE: ERRORS WILL BE SHOWN IN RED!
3. 3. Our Distraction Questions 1. Given the equation 169x2 + 144y2 + 676x – 2016y – 16604 = 0 a) Change to standard form b) Sketch the ellipse c) Find the distance from the point (26, 42) to the foci. 2. Given the equations y = - (x – 3)2 + 4 and y = (x - 2)2 + 1 a) Sketch the graphs b) Find the points of intersection
4. 4. Change 169x2 + 144y2 + 676x – 2016y – 16604 = 0 to standard form
5. 5. Graph the ellipse formed by the equation
6. 6. Find the distance from the point (26, 42) to the foci.
7. 7. Sketch the graphs of y = - (x – 3)2 + 4 and y = (x - 2)2 + 1.
8. 8. Find the points of intersection TRICK QUESTION!! According to our graph there are no points of intersection!
9. 9. The Correct Answers • The mistakes we made in our answer distracted Mr. K long enough for us to steal his coffee, just as we had hoped! • Keep reading the slides to find the correct answers that Mr. K wrote on the smart board!
10. 10. Changing to Standard Form 1. This is the equation 2. Group the X’s and Y’s together , and add 16604 to both sides. The blank space is for completing the square. 3. Factor out 169 from the X terms and 144 from the Y. Complete the square by dividing the middle term (4x, 14y) by two then squaring. Make sure you add this term to the other side! 4. Divide both sides by 169 and 144. 5. Hooray! Now it’s in standard form! :D
11. 11. (h, k) Vertex of the ellipse. Remember to take the opposite sign of what is shown! Half the length of the minor axis. To find the b length of the minor axis multiply b by 2. Half the length of the major axis. To find the a length of the major axis multiply a by 2. To find the foci The foci are c units away from the center. Remember b2 + c2 = a2 so c2 = a2 - b2 To find c
12. 12. Let’s figure out what we know from this equation! Vertex: (-2, 7) We look at h and k to figure this out. Length of a: 13 Take the root of a (169) Length of b: 12 Take the root of b (144) Length of C: 5 c2 = a2 - b2 so c2 = 144 - 169 c2 = 25 c =5 Length of major axis: 26 This is 2a (2 x 13) Length of minor axis: 24 This is 2b (2 x 12) It is a vertical parabola! We know this because a is under the (y – k)2 term.
13. 13. Here are the parts of the graph and where they go
14. 14. Now let’s put it all together!
15. 15. Find the distance from the point (26, 42) to the foci.
16. 16. Sketch the graphs of y = - (x – 3)2 + 4 and y = (x - 2)2 + 1. y = - (x – 3)2 + 4 (x - 2)2 + 1 Vertex: (3, 4) Vertex: (2, 1) The parabola opens The parabola opens up. down. There are no x-intercepts X- intercept: because it’s vertex is 0 = - (x – 3)2 + 4 above the x-axis. = -(x2 -6x + 9) + 4 y- intercept: = -x2 + 6x – 5 y = (0 – 2)2 + 1 = - (x2 – 6x +5) = (4) + 1 = - (x -1)(x - 5) = 2 +1 x = 1, 5 =3
17. 17. Now Let’s Sketch! Start by plotting the vertex, x-intercepts and y- intercepts. Then connect the dots!
18. 18. Find The points of Intersection First lets find them graphically! By looking at our graph we can see the points of intersection are at (4, 3) and approximately (1.3, 1.2)
19. 19. Now find them algebraically. • To find the points of intersection algebraically allow thing 1 and thing 2 to equal each other. • In this case thing 1 is y = - (x – 3)2 + 4 and thing 2 is y = (x - 2)2 + 1 So we would go… –(x – 3) 2 + 4 = (x - 2)2 + 1
20. 20. Remember BEDMAS when solving!! –(x – 3) 2 + 4 = (x - 2)2 + 1 1. Our equation. Solve for x! 2. Expand the x2 terms. -(x2 - 6x +9) + 4 = (x2 - 4x+4) +1 3. Multiply both sides by 2, to -2(x2 - 6x +9) + 8= (x2 - 4x+4) +2 get rid of the denominator. -2x2 +12x -18 + 8 = x2 -4x + 6 4. Distribute the -2 through the brackets. 5. Gather like terms together. 3x2 – 16x + 16 = 0 (3x – 4)(x – 4) = 0 6. Factor! (Remember FOIL) x= x= 4 7. Hooray! We solved for x!!
21. 21. Now let’s solve for y! Substitute your X value into one of the equations to solve for y! x=4 x= y = - (x – 3)2 + 4 y = - (x – 3)2 + 4 = - (4 – 3) 2 + 4 = - ( - 3)2 + 4 = - (1)2 + 4 = - ( - )2 + 4 = -1 + 4 = - ( )2 + 4 =3 = - 2.77778 + 4 = 1.2222 The intercepts are at (4, 3) and (1.3333, 1.2222)
22. 22. In Summary! • We were able to distract Mr. K for long enough to steal his coffee!