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8 Cards in this Set
- Front
- Back
properties of absolute values
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(i) |a|=|-a|
(ii) |a|=0 if and only if a=0 (iii) |ab|=|a||b| (iv) |a/b|=|a|/|b|, b≠0 (v) |a+b|≤|a|+|b| (triangle inequality) (vi) |x|<a if and only if -a<x<a (vii) |x|>a if and only if x>a or x<-a |
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distance between two numbers
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d(a,b)=|b-a|
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midpoint
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m=(a+b)/2
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factorizations worth knowing
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difference of two squares:
a²-b²=(a-b)(a+b) difference of two cubes: a³-b³=(a-b)(a²+ab+b²) sum of two cubes: a³+b³=(a+b)(a²-ab+b²) |
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binomial expansions worth knowing
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n=2: (a+b)²=a²+2ab+b²
n=3: (a+b)³=a³+3a²b+3ab²+b³ |
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vertical and horizontal shifts
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suppose y=f(x) is a function and c is a positive constant.
(i) y=f(x)+c shifts vertically up c units; (ii) y=f(x)-c shifts vertically down c units; (iii) y=f(x+c) shifts horizontally to the left c units; (iv) y=f(x-c) shifts horizontally to the right c units. |
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reflections
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(i) y = -f(x) is the graph of f reflected in the x-axis;
(ii) y =f(-x) is the graph reflected in the y-axis. |
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vertical stretches and compressions
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suppose y = f(x) is a function and c is a positive constant. Then the graph of y = cf(x) is the graph of f
(i) vertically stretched by a factor of c units if c>1; (ii) vertically compressed by a factor of c units if 0<c<1. |