Fractals: Get Stronger Answer Key

  1. Step 2 and Step 3 of a fractal forming a pyramid of right angles
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  3. Step 2 and Step 3 of a fractal forming a branching lines
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  5. Step 2 and Step 3 of a fractal made up of squares with an empty place in the middle
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  9. Four copies of the Koch curve are needed to create a curve scaled by [latex]3[/latex]
    [latex]𝐷=log(4)/log⁡(3)≈1.262[/latex]
    Step 1 and 3 of a fractal
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  11. Eight copies of the shape are needed to make a copy scaled by [latex]3[/latex]. [latex]D=\frac{\log (8)}{\log (3)} \approx 1.893[/latex]

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  13. A graph with the y-axis as imaginary and the x-axis as real. There is a point at 4, 0, labeled 4, a point at 2, 1, labeled 2 + i, a point at 0, 3, labeled -3i, and a point at -2, 3, labeled -2 + 3i.
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    1. [latex]5-i[/latex]
    2. [latex]5-4i[/latex]
    1. [latex]3-5i[/latex]
    2.  [latex]-(6+i)[/latex]
    1. [latex]6+12i[/latex]
    2. [latex]10-2i[/latex]
    3. [latex]14+2i[/latex]
    1. [latex]-2+6i[/latex]
    2. [latex]18+6i[/latex]
    3. [latex]7+3i[/latex]
  19. [latex](2+3i)(1-i) = 5+i.[/latex]It appears that multiplying by [latex]1-i[/latex] both scaled the number away from the origin, and rotated it clockwise about [latex]45°[/latex].A graph with the y-axis labeled imaginary and the x-axis labeled real. There are red dotted lines connected to each of the two points on the graph. There is a point at 5, 1, labeled 5 + i and another point at 2, 3, labeled 2 + 3i.
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  21. [latex]z_{1}=i z_{0}+1=i(2)+1=1+2 i[/latex]
    [latex]z_{2}=i z_{1}+1=i(1+2 i)+1=i-2+1=-1+i[/latex]
    [latex]z_{3}=i z_{2}+1=i(-1+i)+1=-i-1+1=-i[/latex]
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  23. [latex]z_{0}=0[/latex]
    [latex]z_{1}=z_{0}^{2}-0.25=0-0.25=-0.25[/latex]
    [latex]z_{2}=z_{1}^{2}-0.25=(-0.25)^{2}-0.25=-0.1875[/latex]
    [latex]z_{3}=z_{2}^{2}-0.25=(-0.1875)^{2}-0.25=-0.21484[/latex]
    [latex]z_{4}=z_{3}^{2}-0.25=(-0.21484)^{2}-0.25=-0.20384[/latex]
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  25. attracted, to approximately [latex]-0.3776+0.14242i[/latex]
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  27. periodic [latex]2[/latex]-cycle
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  29. Escaping
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  31. periodic [latex]3[/latex]-cycle