San Jacinto College



Pre-Cal Boot Camp: Student PacketParent FunctionsParent FunctionGraphParent FunctionGraphLineary=xDomain: (-∞,∞)Range: (-∞,∞)Symmetry: Odd OriginAbsolute Valuey=IxIDomain: (-∞,∞)Range: [0,∞)Symmetry: Even Y-axisQuadraticy=x2Domain: (-∞,∞)Range: [0,∞)Symmetry: Even Y-axisRadicaly=xDomain: [0,∞)Range: [0,∞)Symmetry: NeitherCubicy=x3Domain: (-∞,∞)Range: (-∞,∞)Symmetry: Odd OriginCube Rooty=3xDomain: (-∞,∞)Range: (-∞,∞)Symmetry: Odd OriginExponentialy=bx, b>1Domain: (-∞,∞)Range: (0,∞)Symmetry: NeitherLogy=logbx, b>1, x>0Domain: (0,∞)Range: (-∞,∞)Symmetry: NeitherRational(Inverse)y=1/xDomain: (-∞,0)U(0,∞)Range: (-∞,0)U(0,∞)Symmetry: Odd OriginRational(InverseSquared)y=1/ x2Domain: (-∞,0)U(0,∞)Range: (0,∞)Symmetry: Even Y-axisGreatest Integery=int(x)=[x]Domain: (-∞,∞)Range:{y:yεZ}(integers)Symmetry: NeitherConstanty=C (in this graph y=2)Domain: (-∞,∞)Range: {y: y=C}Symmetry: Even Y-axisTips and Tricks - Transformations of Functionsy=mx+vy=ax-h2+vy=ax-h+vy=ax-h3+vy=a3x-h+vy=ax-h+vy=a?x-h+vy=a2x-h+vy=alog10x-h+vy=ax-h+vy=a(x-h)2+vTips and Tricks – Vertical Asymptotes and Horizontal Asymptotesfx=1x fx=1x-1fx=x2x-1Tips and Tricks – For x and y Intercepts, Vertical Asymptotes, and Horizontal Asymptotesy=x+1y=x2+3y=x3+6y=xy=x-9y=3xy=3x+10y=1xy=1x-1y=1x2y=1x+42y=1xy=1x-8Tips and Tricks – End Behavior and MultiplicityQuadraticy=x2Domain: (-∞,∞)Range: [0,∞)Symmetry: Even Y-axisQuadraticy= -x2Domain: (-∞,∞)Range: [0,-∞)Symmetry: Even Y-axisCubicy=x3Domain: (-∞,∞)Range: (-∞,∞)Symmetry: Odd OriginCubicy= -x3Domain: (-∞,∞)Range: (-∞,∞)Symmetry: Odd OriginMultiplicity (repeated zeros):A factor of (x - a)k, k > 1, yields a repeated zero x = a of multiplicity k.If k is odd, the graph crosses the x-axis at x = a.If k is even, the graph touches the x-axis at x = a.1. y=x2. y=x23. y=xx-14. y=x2x-345. y=x3x-826. y=- x3x-67. y=-x2x+78.y=x2-819. y=x2-6x+910. y=x2+2x-611. y=x3-25x12. y=x3-x2+2x-213. y=x3+27Tips and Tricks – Inverse Functions and the Horizontal Line TestFinding the Inverse of a Function:The equation for the inverse of a function f can be found as follows:Replace f(x) with y in the equation for f(x)Interchange x and ySolve for y. If this equation does not define y as a function of x, the function f does not have an inverse function and this procedure ends. If this equation does define y as a function of x, the function f has an inverse function.If f has an inverse function, replace y in step 3 by f-1(x). We can verify our result by showing that f(f-1(x))=x REMEMBER: The Horizontal Line Test and One-to-One Functions:A function f has an inverse that is a function, f-1, if there is no horizontal line that intersects the graph of the function f at more than one point.fx=7x-5fx=4x3-1fx=x2fx=5x+4Tips and Tricks – Odd and Even FunctionsDefinition of Even and Odd Functions:The function f is an even function iff(-x)=f(x) for all x in the domain of f.The right side of the equation of an even function does not change if x is replaced with -x.The function f is an odd function iff(-x)=-f(x) for all x in the domain of f.Every term in the right side of the equation of an odd function changes its sign if x is replaced with -x.fx=x3-xfx=x4-x2fx=x2+7fx=x3+4Tips and Tricks – Determining the Following for GraphsUse the graph of f to determine each of the following. Where applicable, use interval notation:The domain of fThe range of fThe x-interceptsThe y-interceptIntervals on which f is increasingIntervals on which f is decreasingIntervals on which f is constantThe number at which f has a relative minimumThe relative minimum of ff(-3)the values of x for which f(x)=-2Is f even, odd, or neither?Tips and Tricks – Difference QuotientDefinition of a Difference Quotient:The expressionfx+h-fxhFor h ≠ 0 is called the difference quotient.For fx=4x , find fx+h-fxhFor fx=x2-4x+7 , find fx+h-fxhFor fx=1x-2 , find fx+h-fxhFor f(x)=x +8, find fx+h-fxhTips and Tricks -For Deriving Pythagorean Identitiessin2θ+cos2θ=1 tan2θ=sec2θ-1cot2θ=csc2θ-1Tips and Tricks – For Even and Odd Trig Identitiescos-θ=cosθsec-θ=secθsin-θ=-sinθcsc-θ=-cscθtan-θ=-tanθcot-θ=-cotθ-54292550355500Trigonometric Parent Functions-61912521844000-62865028575000Blank Unit CircleTips and Tricks – Do’s and Don’ts of Calculator:The Do’s:Texas Instruments TI - 84 Plus CE: Casio FX-115 ES Plus: Best Software Calculator!!!TI SmartView CE for the TI – 84 Plus Family:The Don’ts:Texas Instruments TI-83 series calculatorsTexas Instruments TI-30 series calculatorsTexas Instruments TI-nSpire calculatorsCasio FX-9750 GII calculatorCasio G series calculators-90487531623000Formula Trig Sheet ................
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