Tutor profile: Diana J.
How do you find the X-intercepts, holes, vertical asymptotes, horizontal asymptotes and slant (oblique) asymptotes of a rational function?
1) X-intercepts: Find the values of x for which the numerator of the function becomes 0 2) Holes: Find the factors which occur both in the numerator and in the denominator. They represent the holes. 3) Vertical asymptotes: Find the values of x for which the denominator of the function (except those identified as holes) becomes 0 4) Horizontal asymptotes: If the degree of numerator < degree of denominator, then X-axis itself is the horizontal asymptote. If the degree of numerator = degree of denominator, then y=(leading coefficient of numerator)/(leading coefficient of denominator) is the horizontal asymptote 5) Slant (Oblique) asymptotes: If the degree of numerator > degree of denominator, then y=quotient obtained by dividing numerator by denominator will be the slant (oblique) asymptote
How do you locate and categorize the stationary points of a function using differentiation?
Stationary points are those points of a function where the function is neither increasing, nor decreasing. Those points are located by the property that the derivative will be zero. Stationary points are two types: Turning points and Points of Inflection. Turning points are those where the function changes its increasing/decreasing behavior. Turning points where the function changes from increasing to decreasing are called Points of Maximum. Turning points where the function changes from decreasing to increasing are called Points of Minimum. Points of inflection are those points where 1) either the function is stationary (first derivative = 0) but which does not result in any change in the increasing/decreasing behavior (second derivative = 0) 2) or the function is not stationary (first derivative <> 0) but the second derivative = 0
What are the different forms of a quadratic equation (parabola)? What immediate information do each of them offer us? How can we convert from one form of parabola to another form?
A quadratic equation (parabola) can be written in three different forms as below: 1) General form: y=ax^2+bx+c. This directly gives us the Y-intercept of the parabola which is y=c. This form exists for all parabolas. It can be converted to other forms of parabola by methods like completing the square or factorization. 2) Vertex form: y=a(x-h)^2+k. This directly gives us the vertex of the parabola which is V=(h,k). This form exists for all parabolas. It can be converted to other forms of parabola by methods like binomial expansion and simplification and then factorization if required. 3) Factored form or Intercept form: y=a(x-p)(x-q). This directly gives us the X-intercepts of the parabola which are x=p and x=q. This form exists only for those parabolas which touches or crosses the X-axis. If the parabola just touches the X-axis at a single point, then we get p=q and the equation changes to y=(x-p)^2 It can be converted to other forms of parabola by methods like distributive property (FOIL method) and simplification and then factorization if required.
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