The Coffee Stealing Question

Picture by flickr user Ahmed Rabea

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!

Our Distraction Questions

1. Given the equation 169x2 + 144y2 + 676x – 2016y – 16604 = 0 a) Change to standard form b) Sketch the ellipsec) 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 + 1a) Sketch the graphsb) Find the points of intersection

Change 169x2 + 144y2 + 676x – 2016y – 16604 = 0 to standard form

Graph the ellipse formed by theequation

Find the distance from the point (26, 42) to the foci.

Sketch the graphs of y = - (x – 3)2 + 4 and y = (x - 2)2 + 1.

Find the points of intersection

TRICK QUESTION!!

According to our graph there are no points of intersection!

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!

Changing to Standard Form

1. This is the equation2. 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

(h, k)

b

a

To find the foci

To find c

Vertex of the ellipse. Remember to take the opposite sign of what is shown!

Half the length of the minor axis. To find the length of the minor axis multiply b by 2.

Half the length of the major axis. To find the length of the major axis multiply a by 2.

The foci are c units away from the center.

Remember b2 + c2 = a2 so c2 = a2 - b2

Vertex: (-2, 7)Length of a: 13Length of b: 12Length of C: 5

Length of major axis: 26Length of minor axis: 24

It is a vertical parabola!

We look at h and k to figure this out.

Let’s figure out what we know from this equation!

Take the root of a (169)

Take the root of b (144)

c2 = a2 - b2 so c2 = 144 - 169 c2 = 25 c = 5

This is 2a (2 x 13)

This is 2b (2 x 12)

We know this because a is under the (y – k)2 term.

Here are the parts of the graph and where they go

Now let’s put it all together!

Find the distance from the point (26, 42) to the foci.

Sketch the graphs of y = - (x – 3)2 + 4 and y = (x - 2)2 + 1.

y = - (x – 3)2 + 4Vertex: (3, 4)The parabola opens down.X- intercept:0 = - (x – 3)2 + 4

= -(x2 -6x + 9) + 4= -x2 + 6x – 5= - (x2 – 6x +5)= - (x -1)(x - 5) x = 1, 5

(x - 2)2 + 1Vertex: (2, 1)The parabola opens up. There are no x-intercepts because it’s vertex is above the x-axis. y- intercept: y = (0 – 2)2 + 1 = (4) + 1 = 2 +1 = 3

Now Let’s Sketch!

Start by plotting the vertex, x-intercepts and y- intercepts. Then connect the dots!

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)

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

Remember BEDMAS when solving!!

–(x – 3) 2 + 4 = (x - 2)2 + 1 -(x2 - 6x +9) + 4 = (x2 - 4x+4) +1

-2(x2 - 6x +9) + 8= (x2 - 4x+4) +2

-2x2 +12x -18 + 8 = x2 -4x + 6

3x2 – 16x + 16 = 0(3x – 4)(x – 4) = 0

1. Our equation. Solve for x!

2. Expand the x2 terms.

3. Multiply both sides by 2, to get rid of the denominator.

4. Distribute the -2 through the brackets.

5. Gather like terms together.

6. Factor! (Remember FOIL)

7. Hooray! We solved for x!! x= x= 4

Now let’s solve for y!

Substitute your X value into one of the equations to solve for y!

x = 4y = - (x – 3)2 + 4 = - (4 – 3) 2 + 4 = - (1)2 + 4 = -1 + 4 = 3

x =y = - (x – 3)2 + 4 = - ( - 3)2 + 4 = - ( - )2 + 4 = - ( )2 + 4 = - 2.77778 + 4 = 1.2222

The intercepts are at (4, 3) and (1.3333, 1.2222)

In Summary!

• We were able to distract Mr. K for long enough to steal his coffee!

Mission Accomplished!!