For the equation with a vertex at (3,3): y = -3/4(x-1)(x-5)
For the second equation with a vertex at (5,3): y = -1/3(x-2)(x-8)
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the equation with the vertex (3,3) is y=(-3/4)(x-1)(x-5)
the equation with the vertex (5,3) is y=(-1/3)(x-2)(x-8)
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Ok this is what I have :
3 = a(5-2)(5-8)
a =-1/3
p = 2
q = 8
If you don’t use it, you lose. I hope I can use this in my coding. 
Challenge :
y = -3/4(x-1)(x-5)
Extra :
y = -1/3(x-2)(x-8)
Challenge answer:
0 = -1/3 (x - 2) (x - 8)
How I solved it:
3 = a(5 - 2) (5 - 8)
3 = a * (3) * (-3)
3 = a * -9
3 / -9 = a
a = -1/3
I did a lot of scribbling around but got there in the end!
Challenge answer:
y = -3/4(x-1)(x-5)
Bonus Challenge:
y = -1/3(x-2)(x-8)
Hello there !
Here is my answer to both challenges
- For root1 (1,0); root2 (5,0); height (3,3)
if y=o,
0 = a (x - 1)(x - 5)
if y = 3 and x = 3
3 = a (3 - 1)(3 - 5)
3 = a (2)(-2)
3 = - 4a
a = - 3/4
So
[ 0 = -3/4 (x - 1)(x - 5) ]
- For root1 (2,0); root2 (8,0); height (5,3)
if y=o,
0 = a (x - 2)(x - 8)
if y = 3 and x = 5
3 = a (5 - 2)(5 - 8)
3 = a (3)(-3)
3 = - 9a
a = - 3/9
a = - 1/3
So
[ 0 = -1/3 (x - 2)(x - 8) ]
Cheers
Edit: I thought the spoiler was automatic 
Just figured how to do it
Original challenge: 0=-0.75(x-1)(x-5)
Updated challenge: 0=(-1/3)(x-2)(x-8)
It took some scrubbing over my notes, but I came up with the following:
y= -3.4(x-1)(x-5)
and
y = -1/3 (x-2)(x-8)
y = a * (x - 2) * (x - 8)
3 = a * (5 - 2) * (5 - 8)
3 = a * 3 * (-3)
a = -1/3
- For the points (1, 0), (5, 0) and (3, 3)
0 = a (x -1) (x - 5)
p = 1, q = 5
3 = a (3 -1) (3 - 5)
3 = a (2) (-2)
a = -3/4 - For the points (2, 0), (8, 0) and (5, 3)
0 = a (x -2) (x - 8)
p = 2, q = 8
3 = a (5 -2) (5 - 8)
3 = a (3) (-3)
a = -3/9
a = -1/3
No clue if I got this right. Had to keep going back and watching the process and tried just plugging my values in place. 
Higher Shot:
3 = a(1-3)(5-3)
3 = a(-2)(2)
3 = -4a
-3/4 = a
[0 = (-3/4)(x-3)(x-3)]
Further Spread:
3 = a(2-5)(8-5)
3 = a(-3)(3)
3 = -9a
-3/9 = a
-1/3 = a
[0 = (-1/3)(x-5)(x-5)]
Here’s my answer and work for the Intercept Form challenge:
Q1. Given the roots (1, 0) and (5, 0) and a vertex of (3, 3), write the intercept form.
A1. 0 = -3/4(x - 1)(x - 5)
A1a. 0 = a(x - 1)(x - 5)
A1b. 3 = a(3 - 1)(3 - 5)
A1c. 3 = a(2)(-2)
A1d. 3 = -4a
A1e. a = -3/4
Q2. Given the roots (2, 0) and (8, 0) and a vertex of (5, 3), write the intercept form.
A2. 0 = -1/3(x - 2)(x - 8)
A2a. 0 = a(x - 2)(x - 8)
A2b. 3 = a(5 - 2)(5 - 8)
A2c. 3 = a(3)(-3)
A2d. 3 = -9a
A2e. a = -3/9
A2f. a = -1/3
Very good, can you simplify the last step?
Done and done - simplified to its simplest!
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The intercept form for curve (2,0) - (5,3) - (8,0) would be:
y = -1/3(x-2)(x-8)
And the one for (1,0) - (3,3) - (5,0) would be:
y = -3/4(x-1)(x-5)
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