Graphs of Rational Functions

1
Graphs of Rational Functions

• 3.4

2
Steps for Graphing a Rational Function

• 1. Find the Domain.
• set the denominator to zero, then all reals
  except those numbers.
• 2. Find the intercepts.
• To find the x intercepts, set y to zero. Y
  will be zero when the numerator is zero, so set
  to zero and solve. To find the y-intercepts, set
  x to zero and solve.

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Steps for Graphing a Rational Function Continued

• 3. Test whether the graph is symmetric with
  respect to the y-axis or origin.
• Originplug in x for x and see if the new
  function is the opposite of the old. Or if (x,y)
  is on graph, then (-x,-y) is on graph.
• Y-axisplug in x for x and see if the new
  function is the same as the old. Or if (x,y) is
  on the graph, then (-x,y) is on the graph.
• 4. Find the vertical asymptotes.
• Numbers that make the denominator zero, but not
  the numerator.

4
Steps for Graphing a Rational Function Continued

• 5. Find the horizontal and oblique asymptotes.
• If the degree of the numerator the degree of
  the denominator, then the HA is the coefficients
  of the highest degree terms.
• If the degree of the numerator is lower than
  the degree of the denominator, then the HA is
  zero.
• If the degree of the numerator is higher than
  the degree of the denominator, then there is no
  HA.
• 6. Find the oblique asymptotes.
• Occur only when the degree of the numerator
  is one more than the degree of the denominator.

5
Steps for Graphing a Rational Function Continued

• 7. Graph the function on the calculator.
• 8. Use the results above to graph the equation
  by hand.

6
Polynomial and Rational Inequalities

• 3.5

7
Solving Polynomial Inequalities

• Put all terms on the left side and zero on the
  other.
• Simplify the terms and then factor.
• Draw a number line.
• Plot the zeroes on the number line.

8
Solving Polynomial Inequalities

• This divides the number line up into sections.
  Plug a number from each section into the original
  expression.
• If the answer is positive put a plus sign in that
  section, otherwise put a minus sign.
• Choose the intervals with minus signs if the
  inequality is a less than. Choose the positive
  intervals if the inequality is a greater than.
  Include the zeroes if the less than or greater
  than includes an equal sign.

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Solving Rational Inequalities Algebraically

• Rearrange the terms so that 0 is on the right
  side.
• Get a single fraction on the left, by creating a
  common denominator and adding the algebraic
  fractions.
• Find the zeros of f by setting the numerator
  equal to zero.
• Find the values at which f is undefined by
  setting the denominator equal to zero. (or
  negatives under a square root)

10
Solving Rational Inequalities Algebraically

• Draw a number line and plot the zeros and the
  values that make the function undefined.
• Select a number in each interval and plug the
  number into the original expression. If the
  result is positive, then put pluses in that
  section, otherwise put minuses.

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Solving Rational Inequalities Algebraically

• Choose the negative intervals if the inequality
  is a less than. Choose the positive intervals if
  the inequality is a greater than. Include the
  zeros(not the undefined) if the less than or
  greater than includes an equal sign.

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Solving Rational Inequalities Graphically

• Put the left side of the inequality in y1 making
  sure that the entire numerator and denominator
  are surrounded by parenthesis in a fraction.
• Put the right side in y2.
• Use 2nd Calc Intersect to find where the graphs
  intersect.
• Plot these points on a number line.

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Solving Rational Inequalities Graphically

• Place pluses in the sections where the graph on
  the left is lt, gt,, or (depending on the sign
  in the original problem) the graph on the right.
• Write these sections in interval notation.