the graph of a quadratic function f intersects the x-axis st x= -2 and x=6. If f(8) = f(p), which could be the value of p?
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f(x)= a(x+2)(x-6)
so, f(8)= a*10*2 = 20*a
and,
f(p)=a*(p+2)(p-6)=a*(p^2-4p-12)
f(8)=f(p)
20*a=a*(p^2-4p-12)
20=p^2-4p-12
0=p^2-4p-32
0=(p-8)(p+4)
since p is not 8, p=-4
so, f(8)= a*10*2 = 20*a
and,
f(p)=a*(p+2)(p-6)=a*(p^2-4p-12)
f(8)=f(p)
20*a=a*(p^2-4p-12)
20=p^2-4p-12
0=p^2-4p-32
0=(p-8)(p+4)
since p is not 8, p=-4
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Well, obviously p could be 8. The other value is the reflection of the point (8, f(x)) across the line of symmetry.
The line of symmetry is easy to see as x = 2 given the zeroes at (-2,0) and (6,0). (2,0) is halfway between those points.
So f(2+d) = f(2-d), for any value of d.
That's the symmetry equation, using d mean "distance from the line of symmetry". If 2+d = 8, then p could be either 2+d or 2-d and have f(p) = f(8). But 2+d = 8 means that d=6, so the possible values for p are 2+6=8 and 2-6 = -4.
The line of symmetry is easy to see as x = 2 given the zeroes at (-2,0) and (6,0). (2,0) is halfway between those points.
So f(2+d) = f(2-d), for any value of d.
That's the symmetry equation, using d mean "distance from the line of symmetry". If 2+d = 8, then p could be either 2+d or 2-d and have f(p) = f(8). But 2+d = 8 means that d=6, so the possible values for p are 2+6=8 and 2-6 = -4.
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The equation of the parabola is y = (x+2)(x-6) = x^2-4x-12
when x = 8, y = f(8) = 8^2 - 4x8 - 12 = 64 - 32 - 12 = 20
when x = 8, y = f(8) = 8^2 - 4x8 - 12 = 64 - 32 - 12 = 20