Question 1198745
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Ignore the solution below.<br>
Tutor @ikleyn's comments are correct.  There are an infinite number of quadratic functions that pass through the two given points; however, the given form of the quadratic function "y = a(x-h)^2" is of a function whose graph has its vertex on the x-axis.<br>
There is only one such quadratic function -- the one she shows in her response.<br>
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Actually, there is some good mathematics in my solution....  But it doesn't answer the question that is asked.<br>
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The problem is deficient; there are an infinite number of quadratic functions whose graphs pass through the two given points.<br>
Two points determine a unique straight line (linear function).<br>
Three points are needed to determine a unique parabola (quadratic function).<br>
Two points determine only an infinite family of quadratic functions.<br>
Use the standard form of a quadratic equation<br>
{{{y=ax^2+bx+c}}}<br>
with the given two points to find parametric equations for the coefficients a, b, and c.<br>
(12,-1): 144a+12b+c = -1
(9,0): 81a+9b+c = 0<br>
Subtract one equation from the other to eliminate c:<br>
63a+3b = -1<br>
Solve for b in terms of a:<br>
b = (-63a-1)/3<br>
Substitute that expression for b in either original equation to find c in terms of a:<br>
81a+9((-63a-1)/3)+c = 0
81a-189a-3 + c = 0
c = 108a+3<br>
Use t as a parameter to get parametric equations for a, b, and c.<br>
a = t
b = (-63t-1)/3
c = 108t+3<br>
t = (-1/9) gives the quadratic function shown by the other tutor (which has the given point (9,0) as the vertex):
a = t = -1/9
b = (-63(-1/9)-1)/3 = (7-1)/3 = 2
c = 108(-1/9)+3 = -12+3 = -9
y = (-1/9)x^2+2x-9<br>
Choose a couple of other values for parameter t to find other quadratic functions that pass through the two given points.<br>
t = 1
a = 1
b = (-63-1)/3 = -64/3
c = 108+3 = 111
y = x^2-(64/3)x+111<br>
t = -1
a = -1
b = (63-1)/3 = 62/3
c = -108+4 = -105
y = -x^2+(62/3)x-105<br>
Here are graphs of those three quadratic functions graphed in the same window x from 0 to 20 and y from-10 to 10, showing all three parabolas passing through the two given points (9,0) and (12,-1):<br>
{{{graph(400,400,0,20,-10,10,(-1/9)x^2+2x-9,x^2-(64/3)x+111,(-1)x^2+(62/3)x-105)}}}<br>