Reveiwer: Paul Kates Some useful online plotting tools for checking questions:
Path to reviewed questions:
High School Enrichment > Advanced Functions > 4. Rational Functions > Group: 4.1 Reciprocal of a Polynomial Function
Comments:
Where the original linear function is 0 (the x-intercept or root of the function), its reciprocal function will have a vertical asymptote line and a plot on either side of this line. The linear and reciprocal functions will have the same sign for y on the left side of the root and the left side of the asymptote line. Similarly, the linear and reciprocal functions will have the same sign for y on the right side of the root and the right side of the asymptote line.Question answer:
Comment: When trying to match the graphs (for the case of linear polynomials: f(x) = mx + b ^ extra ( or missing ) Determine the sign of the slope of the linear function - is it a positive or a negative slope? The reciprocal of a linear function with POSITIVE slope will have branches (the two distinct parts of the graph) on the bottom left and top right portions of the graph. The reciprocal of a linear function with NEGATIVE slope will have branches on the bottom right and top left portions of the graph. Look at the x-intercepts of the function; these will be the x-values at which the reciprocal functions will have asymptotes.
Comments:
Question answer:
Comment: When matching the graphs (for the case of quadratic polynomials f(x) = ax^2 + bx +c: ^ extra ( or missing ) Look at the x-intercepts of the function; these will be the x-values at which the reciprocal functions will have asymptotes. For example, See maple image 1 below: 2 intercepts. If there is only one x-intercept, there will be two symmetric branches (distinct parts) to the graph -- they will open upward if the original parabola opened up, and downward if the original parabola opened downward. For example, See maple image 2 below: 1 intercept. If there are no x-intercepts, the reciprocal function will appear as a "bump" along a horizontal asymptote. For example, See maple image 3 below: 0 intercept.
Image 1: 2 intercepts:
Image 2: 1 intercept:
Image 3: 0 intercepts:
In general, the comment section has a lots of cases which is nice to look at and good to include, but likely won't be memorized. So the general answer helps only a little. Maybe also add a description with a focus on the parts of the x-axis where f(x) is +ve and -ve and 0. And then look for the same signs in the reciprocal plot 1/f(x) and visa versa.
Image 1: Last plot of "4. f(x) is linear" but f(x) isn't linear:
Image 2: 1 intercept.
Question answer:
Comment: First, determine where the asymptotes are located, if any exist. ^ in function y=1/f(x) These will be the x-intercepts of f(x), the denominator of the given reciprocal function. Furthermore, you must check the sign of the function: ... ^ f(x)
Paul Kates
Mathematics Faculty CTE Liaison
pkates@uwaterloo.ca, x37047
Last modification date: Thu Jul 10 15:10:16 2014.