College Algebra – Flashcards
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Distance Formula
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d(P,Q)=?[(x??x?)²+(y??y?)²] P (x?,y?) Q (x?,y?)
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Midpoint Formula
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M(x,y)=[(x?+x?)/2, (y?+y?)/2] P(x?,y?) Q(x?,y?)
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X-intercepts
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where graph touches x-axis or crosses set y=0 and solve for x
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Y-intercepts
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where graph touches y-axis or crosses set x=0 and solve for y
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symmetry to y-axis
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(x,y) (?x,y)
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symmetry to x-axis
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(x,y) (x,?y)
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symmetry to origin
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(x,y) (?x,?y)
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circle components
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center of circle (h,k) radius=r
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standard equation of circle
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(x?h)²+(y?k)²=r²
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general form of equation of circle
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x²+y²+ax+by+c=0
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Slope of a Non-Vertical line
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m=rise/run=(y??y?)/(x??x?) P(x?,y?) Q(x?,y?)
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Slope of a Vertical line
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undefined
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Equation of a line- point slope form
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y?y?=m(x?x?)
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Equation of a line-slope intercept form
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y=mx+b b=y intercept
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Equation of a line-horizontal line
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y=k
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Equation of a line-vertical line
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x=k
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Equation of a line-general form
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ax+by+c=0
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Parallel Lines
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two lines with respective slopes m? and m? are parallel if m?=m?
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Perpendicular Lines
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two lines with respective slopes m? and m? are perpendicular if m?m?= ?1 (one will be reciprocal)
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Definition of a Relation
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any set of ordered pairs
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Domain of a set (x,y)
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first component
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Range of a set (x,y)
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second component
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Definition of a Function
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a relation in which no two ordered pairs have the same first component; a special type of relation
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Vertical Line Test
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intersects at no more than one point=graph of a function
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Domain of a Function
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the largest set of real numbers for which f(x) is a real number, if a function f is defined by an equation
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Average Rate of Change
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[f(b)?f(a)]/b?a, b?a (so not equal to 0)
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Difference Quotient
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[f(x+h)?f(x)]/h, h?0
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Set Notation
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{x|x? ?1 and x? 1}
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Interval Notation
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(-?,?1)?(?1, 1)?(1,?) (excludes, [ includes
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Piecewise Function
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f(x)=|x|={?x if x<0 {x if x ?0
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Increasing Function
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rising graph left to right
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Decreasing Function
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falling graph left to right
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Constant Function
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staying the same left to right
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Even Function
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f(?x)=f(x); graph is symmetric to y-axis ex. y=x²
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Odd Function
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f(?x)= ?f(x); graph is symmetric to origin ex. y=x³
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Identity Function
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f(x)=x graph=line D (??,?) R(??,?); odd function
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Constant Function
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f(x)=c graph=horizontal line m=0
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Squaring Function
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f(x)=x² graph=parabola (?) D (??,?) R (y ?0) even function
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Cubing Function
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f(x)=x³ graph= ? through origin across y-axis D (??,?) R (-?,?) odd function
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Absolute Value Function
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f(x)=|x| graph= V D (??,?) R [0,?) even function
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Square Root Function
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f(x)=?(x) graph=? extending (0,?) D (x?0) R (0,?)
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Cube Root Function
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f(x)=³?(x) graph= ? across x-axis D (??,?) R (??,?) odd function
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Reciprocal Function
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f(x)=1/x graph=? in 3rd quadrant and ? in 1st quadrant D (??,?) R (??,?) odd function
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Greatest Integer Function
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f(x)=[|x|] graph= "step function"
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Shifts f(x)+c
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shift up by c units
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Shifts f(x)-c
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shift down by c units
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Shifts f(x-c)
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shift right by c units
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Shifts f(x+c)
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shift left by c units
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reflections -f(x)
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reflect across x-axis
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reflections f(-x)
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reflect across y-axis
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Transformation Steps
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1. Horizontal 2. stretch/compress 3. reflect 4. vertical
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Stretch & Compress
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g(x)=a·f(x) a>0 has the same shape as f(x); taller if a>1 and flatter if 0<a<1
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Stretch & Compress-away from y
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g(x)=f(bx) stretch away from y-axis if 0<b 1
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Sums of Functions
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(f+g)(x)=f(x)+g(x)
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Difference of Functions
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(f-g)(x)=f(x)?g(x)
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Product of Functions
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(fg)(x)=f(x)·g(x)
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Quotient of Functions
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(f/g)(x)=f(x)/g(x) where denominator ? 0
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Compositions of Functions
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(f°g)(x)=f(g(x))
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Domain of f°g
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set of all x in the domain of g that g(x) is in the domain of f; in general f°g ? g°f
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One-to-One Functions
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if for any x? and x? in the domain of f, f(x?)=f(x?)
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Horizontal Line Test
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intersects in at most one point, then f(x) is a one-to-one function
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Inverse Function
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g=f?¹(x) if function is 1:1
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Graph of an Inverse Function
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graph of f?¹ is reflection of graph f over the line y=x
