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RATIONAL

RATIONAL. FUNCTIONS II. GRAPHING RATIONAL FUNCTIONS. Connect points and head towards asymptotes. Find some points on either side of each vertical asymptote. Find horizontal or oblique asymptote by comparing degrees. Test for symmetry by putting – x in for x . (remember even, odd test).

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RATIONAL

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  1. RATIONAL FUNCTIONS II GRAPHING RATIONAL FUNCTIONS

  2. Connect points and head towards asymptotes. Find some points on either side of each vertical asymptote Find horizontal or oblique asymptote by comparing degrees Test for symmetry by putting –x in for x. (remember even, odd test) Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving. Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x Find the domain. Excluded values are where your vertical asymptotes are. Steps to GraphingRational Functions

  3. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Find the domain. Excluded values are where your vertical asymptotes are.

  4. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x So let’s plot the y intercept which is (0, - 1)

  5. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving. 2 If the numerator of a fraction = 0 then the whole fraction = 0 since 0 over anything = 0 But 0 = 6 is not true which means there IS NO x intercept.

  6. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 Test for symmetry by putting –x in for x. (remember even, odd test) 0 4 6 8 2 Not the original and not negative of function so neither even nor odd.

  7. Find horizontal or oblique asymptote by comparing degrees -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 degree of the top = 0 remember x0 = 1 degree of the bottom = 2 If the degree of the top is less than the degree of the bottom the x axis is a horizontal asymptote.

  8. Choose an x on the right side of the vertical asymptote. Find some points on either side of each vertical asymptote -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Choose an x on the left side of the vertical asymptote. x R(x) -4 0.4 1 -1 4 1 Choose an x in between the vertical asymptotes.

  9. Pass through the point and head towards asymptotes Connect points and head towards asymptotes. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Pass through the point and head towards asymptotes There should be a piece of the graph on each side of the vertical asymptotes. Go to a function grapher or your graphing calculator and see how we did. Pass through the points and head towards asymptotes. Can’t go up or it would cross the x axis and there are no x intercepts there.

  10. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 The window on the calculator was set from -8 to 8 on both x and y. Notice the calculator draws in part of the asymptotes, but these ARE NOT part of the graph. Remember they are sketching aids---the lines that the graph heads towards.

  11. Let's try another with a bit of a "twist": Find the domain. Excluded values are where your vertical asymptotes are. vertical asymptote from this factor only since other factor cancelled. But notice that the top of the fraction will factor and the fraction can then be reduced. We will not then have a vertical asymptote at x = -3, but it is still an excluded value NOT in the domain.

  12. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Find the y intercept if there is one. Remember we find the y intercept by putting 0 in for x We'll graph the reduced fraction but we must keep in mind that x - 3 So let’s plot the y intercept which is (0, - 1/3)

  13. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 Find the x intercepts if there are any by setting the numerator of the fraction = 0 and solving. 2 If the numerator of a fraction = 0 then the whole fraction = 0 since 0 over anything = 0 x + 1 = 0 when x = -1 so there is an x intercept at the point (-1, 0)

  14. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 Test for symmetry by putting –x in for x. (remember even, odd test) 0 4 6 8 2 Not the original and not negative of function so neither even nor odd.

  15. Find horizontal or oblique asymptote by comparing degrees -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 degree of the top = 1 1 1 degree of the bottom = 1 If the degree of the top equals the degree of the bottom then there is a horizontal asymptote at y = leading coefficient of top over leading coefficient of bottom.

  16. We already have some points on the left side of the vertical asymptote so we can see where the function goes there Find some points on either side of each vertical asymptote -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 x S(x) 4 5 6 2.3 Let's choose a couple of x's on the right side of the vertical asymptote.

  17. Pass through the point and head towards asymptotes Connect points and head towards asymptotes. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Pass through the points and head towards asymptotes There should be a piece of the graph on each side of the vertical asymptote. REMEMBER that x -3 so find the point on the graph where x is -3 and make a "hole" there since it is an excluded value. Go to a function grapher or your graphing calculator and see how we did.

  18. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 The window on the calculator was set from -8 to 8 on both x and y. Notice the calculator drew the vertical asymptote but it did NOT show the "hole" in the graph. It did not draw the horizontal asymptote but you can see where it would be at y = 1.

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