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Topic 2: Functions and EquationsPolynomialsThe Remainder Theorem states: The Factor Theorem states:If a polynomial f(x) is divided by x-k, thenremainder=f(k)A polynomial f(x) has a factor (x-k) if and only if : fk=0Polynomial function: Factors, Roots, Zeros y=x2+2x-15Factors are: (x+5) and (x-3)The line of symmetry of y=ax2+bx-c is: x=-b2aThis can also be used to find turning point of quadratic by plugging xZeros are: -5 and 3X-Intercepts are at: -5 or -3The number of solutions of a quadratic equation depends on the value of the discriminant:?=b2-4acRoots/Solutions are: x=5 or 3?>02 Real distinct solutions?=0One Real Solution?<0No real solutionsTopic 2: Functions and EquationsThe Theory of FunctionsFunction: A set of ordered pairs in which every x-value has a unique y-value.In order to be a function, the graph of an equation must pass the vertical and horizontal line testThe Vertical Line Test States:A relation is a function if a vertical line intersects the graph of a relation at only one point, The Horizontal Line Test States:A function is a one-to-one function if a horizontal line crosses the graph onceOtherwise, it is a many-to-one functionRationale Functions are a ratio of two polynomials:Asymptote & intercepts of a rational function:Vertical Asymptote: VA=-dc (where y is impossible, thus denominator=0)Horizontal Asymptote: HAdegnum=deg?(den)→=ac (substitute ∞ for x)degnum<deg?(den)→=0degnum>deg?(den)→=nonefx=ax+bcx+dX-intercept: x=-ba (where y=0)Y-intercept: y=bd (where x=0)Interval NotationSet Builder NotationA function is odd when: f-x=-f(x)-1206500-17907000160020-20129500A function is even when: f-x=f(x)Inverse functions:f-1(x)Reflection of f(x) on the line y=xSwaps domain and range of f(x)f(f-1(x))=f(x)Topic 2: Functions and EquationsTransformations of GraphsShiftsy=fx-h shifts y=f(x) to the right by h unitsy=fx+h shifts y=f(x) to the left by h unitsy=fx+k shifts y=f(x) up by h unitsy=fx-k shifts y=f(x) down by h unitsReflectionsy=f-x reflects y=f(x) across the y-axisy=-fx reflects y=f(x) across the x-axisStretchesIf a>1, transformation is a stretchIf a<1, transformation is a compressy=fax stretches/compresses y=f(x) horizontally, by 1ay=afx stretches/compresses y=f(x) vertically, by aModulusf(x)Turns all x values positivef(x)Reflects the graph to the right of the y-axis in the y-axisIgnore the left hand side part of the graph1f(x)Zeros of f(x) (when they exist) are the vertical asymptotes of 1f(x)Zeros of 1f(x) are the vertical asymptotes of f(x)If c the y-intercept of f(x), then 1c is the y-intercept of 1f(x)The minimum value of f(x) is the maximum of 1f(x)The minimum value of 1f(x) is the maximum of f(x)When fx>0, 1f(x)>0When fx<0, 1f(x)<0When f(x) approaches 0, 1f(x) will approach ±∞When f(x) approaches ±∞, 1f(x) approaches 0 ................
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