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You can put this solution on YOUR website!
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\n" ); document.write( "Ignore the solution below.
\n" ); document.write( "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.
\n" ); document.write( "There is only one such quadratic function -- the one she shows in her response.
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\n" ); document.write( "Actually, there is some good mathematics in my solution.... But it doesn't answer the question that is asked.
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\n" ); document.write( "The problem is deficient; there are an infinite number of quadratic functions whose graphs pass through the two given points.
\n" ); document.write( "Two points determine a unique straight line (linear function).
\n" ); document.write( "Three points are needed to determine a unique parabola (quadratic function).
\n" ); document.write( "Two points determine only an infinite family of quadratic functions.
\n" ); document.write( "Use the standard form of a quadratic equation
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\n" ); document.write( "with the given two points to find parametric equations for the coefficients a, b, and c.
\n" ); document.write( "(12,-1): 144a+12b+c = -1
\n" ); document.write( "(9,0): 81a+9b+c = 0
\n" ); document.write( "Subtract one equation from the other to eliminate c:
\n" ); document.write( "63a+3b = -1
\n" ); document.write( "Solve for b in terms of a:
\n" ); document.write( "b = (-63a-1)/3
\n" ); document.write( "Substitute that expression for b in either original equation to find c in terms of a:
\n" ); document.write( "81a+9((-63a-1)/3)+c = 0
\n" ); document.write( "81a-189a-3 + c = 0
\n" ); document.write( "c = 108a+3
\n" ); document.write( "Use t as a parameter to get parametric equations for a, b, and c.
\n" ); document.write( "a = t
\n" ); document.write( "b = (-63t-1)/3
\n" ); document.write( "c = 108t+3
\n" ); document.write( "t = (-1/9) gives the quadratic function shown by the other tutor (which has the given point (9,0) as the vertex):
\n" ); document.write( "a = t = -1/9
\n" ); document.write( "b = (-63(-1/9)-1)/3 = (7-1)/3 = 2
\n" ); document.write( "c = 108(-1/9)+3 = -12+3 = -9
\n" ); document.write( "y = (-1/9)x^2+2x-9
\n" ); document.write( "Choose a couple of other values for parameter t to find other quadratic functions that pass through the two given points.
\n" ); document.write( "t = 1
\n" ); document.write( "a = 1
\n" ); document.write( "b = (-63-1)/3 = -64/3
\n" ); document.write( "c = 108+3 = 111
\n" ); document.write( "y = x^2-(64/3)x+111
\n" ); document.write( "t = -1
\n" ); document.write( "a = -1
\n" ); document.write( "b = (63-1)/3 = 62/3
\n" ); document.write( "c = -108+4 = -105
\n" ); document.write( "y = -x^2+(62/3)x-105
\n" ); document.write( "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):
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