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Parabola_TangLines_Activity
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(4) Use interval notation and inequalities to describe all the slopes for -5≤a≤5. Do the same for the y-intercepts.
(4) Use interval notation and inequalities to describe all the slopes for -5≤a≤5. Do the same for the y-intercepts.
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(5) What is the minimum absolute value of the slopes of these lines? At what point on the parabola does this occur? What is this point called?
(5) What is the minimum absolute value of the slopes of these lines? At what point on the parabola does this occur? What is this point called?
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(6) Solve the system y=x²,y=2ax-a² to show that, in terms of 'a', the point of intersection of each line and the parabola is (a,a²).
(6) Solve the system y=x²,y=2ax-a² to show that, in terms of 'a', the point of intersection of each line and the parabola is (a,a²).
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For Teacher
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There are many Common Core objectives here but my primary goal is to provide a simple demo of how to use Desmos to create an interactive exploration.
There are many Common Core objectives here but my primary goal is to provide a simple demo of how to use Desmos to create an interactive exploration.
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This Investigation can be significantly improved with Desmos' New Activity Builder!
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Answers
Answers
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(1) x²=6x-9 --> x²-6x+9=0, (x-3)²=0. x=3 is the only solution which shows that (3,9) is the only point of intersection.
(1) x²=6x-9 --> x²-6x+9=0, (x-3)²=0. x=3 is the only solution which shows that (3,9) is the only point of intersection.
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(2) Similar to a tangent line to a circle, y=6x-9 has exactly 1 point of intersection with the curve.
(2) Similar to a tangent line to a circle, y=6x-9 has exactly 1 point of intersection with the curve.
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(3) Slope=2a; y-int= -a²
(3) Slope=2a; y-int= -a²
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(4) -10≤slope≤10 or [-10,10]; -25≤y-int≤0 or [-25,0]
(4) -10≤slope≤10 or [-10,10]; -25≤y-int≤0 or [-25,0]
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(5) 0; (0,0); Vertex
(5) 0; (0,0); Vertex
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(6) x²=2ax-a² --> x²-2ax+a²=0 --> (x-a)²=0 -->x=a --> y=a²
(6) x²=2ax-a² --> x²-2ax+a²=0 --> (x-a)²=0 -->x=a --> y=a²
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