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Substitute all the x's in the reduced function by 0 and solve. Write as (0 ,__) and plot this point on the graph. If you end up with a zero on the as the denominator then that means that there is no y-intercept.
Nominator
Denominator
Does this factor match a factor in the numerator?
Set functions in the numerator from the reduced function equal to zero and solve for x. Write as (__ , 0). These are your x-intercepts, the points on the graph that the function goes through.
Yes
No
POD
Vertical Asymptote
To find POD take out the factors that are both in the numerator and the denominator.This will give you your reduced function (use this function for the rest of the flowchart). Set the factor that you took out equal to zero, solve for x, and write as x ≠ #. Plug in that # in your reduced function which will give you a y value. Write as ( #, y) and draw an open circle at that point. ( Your function cannot cross this point ! )
Find asymptote by setting factor equal to zero and solving for x. Draw a dotted line at x=#
Make sure the numerator and denominator are arranged in descending order of power to find horizontal/slant asymptotes.
To determine the behavior of the function from both sides of the asymptote you have to plug in a number slightly more negative than the value that the asymptote is on to determine the behavior on the left of the asymptote and slightly more positive to determine the behavior on the right of the asymptote into the reduced function. If the outcome is positive, the function will go up. If the out come is negative, the function will go down.
To determine if the function will be above or below the asymptote on the left and the right side of the graph you have to plug in a really large number into the function. If you you want to determine the end behavior on the left of the graph, plug in a really large negative number. If you want to find the end behavior on the right of the graph plug in a really large positive number. If the outcome is larger than the asymptote then the function is above it, and if it is smaller than the asymptote then it is below it.
If the degree of the leading term of a(x) < the degree of the leading term of b(x) then . . .
. . . there is a horizontal asymptote.Draw a dotted line at y=0
If the degree of the leading term of a(x) is = the degree of the leading term of b(x) then . . .
. . . there is a horizontal asymptote. Draw a dotted line at y = the leading coefficient of a(x) / the leading coefficient of b(x).
To determine the end behavior of the function plug in a a large number into the equation with the remainder and into the equation without the remainder and compare. To find out the behavior on the left of the graph plug in a large negative number and to find the behavior on the right plug in a large positive number. If the result of the equation with the remainder is bigger than the result of the equation without the remainder then the function is above the asymptote, if it is less then below the asymptote.
If the degree of the leading term of a(x) is exactly one more than the degree of the leading term of b(x) then . . .
. . . the graph will have a slant asymptote. To find the slant asymptote use long division to solve the reduced function and find the equation. The line for the slant asymptote will be the solution to the long division. The solution will most likely have a remainder. Use that equation to draw a dotted line.
If the degree of the leading term of a(x) is > than the leading term of b(x) by more than one then there is no asymptote.