Given zero, find other zeros. Parabola Writing Equations given zeros Inequalities Write Equation Given a Sketch Word Problem Intermediate Value Theorem \ Bounds Rational Root Theorem Polynomial Information Complex Numbers Rational Functions\Asymptotes Click on buttons to go to a topic. Not all topics are covered here, but this should help you study. Click home buttons on the bottom right of each page to come back to this screen. If there is an error or a question, please notify me by e-mail or AIM.
Transcript
Given zero, find other zeros.
Parabola
Writing Equations given zeros
Inequalities
Write Equation Given a Sketch
Word Problem
Intermediate Value Theorem \ Bounds
Rational Root Theorem
Polynomial Information
Complex Numbers
Rational Functions\Asymptotes
Click on buttons to go to a topic.
Not all topics are covered here, but this should help you study.
Click home buttons on the bottom right of each page to come back to this screen.
If there is an error or a question, please notify me by e-mail or AIM.
Given zeros, find other zeros.
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1) Use synthetic to find the depressed equation given a zero.
2) If first zero was complex, use the conjugate to find the depressed equation again.
3) Reduce until you reach a quadratic, then factor or use quadratic formula to find the other zeros.
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1) Write the equation of the formula in vertex form using completing the square!
2) State the vertex and the axis of symmetry
3) Which way does it open and why?
4) State the intercepts
5) Describe the transformation and graph, state all key points
6) State the range
7) State the intervals of increase and decrease
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Remember to balance the equation.
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Use parenthesis.
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Intermediate value theorem, bounds.
Intermediate value theorem: Given a continuous function in the interval [a,b], if f(a) and f(b) are of different signs, then there is at least one zero between a and b.
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f(3) = -9
f(4) = -7
f(5) = -3
f(6) = 3
There is a zero in the interval [5,6] because there is a sign change, and by intermediate value theorem, a zero must exist in that interval.
Regarding bounds, just understand the application of the formula and what it means. Remember, in using the formula, you don’t use the leading coefficient.
Bounds. Let f denote a polynomial function whose leading coefficient is 1. A bound M on the zeros of f
Inequalities 1) Move Polynomial so that f is on the left side, and zero is on the right side. Write as a single quotient (Common denominator)
2) Determine the numbers where f equals zero or is undefined.
3) Use those values to separate the real number line. (open and closed)
4) Select a number in each interval and evaluate.
1) If f(x) > 0, all x’s in interval are greater than zero.
2) If f(x) < 0, all x’s in interval are less than zero.
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Open, even though it’s equals to, it’s undefined there, so x can’t equal 1 or -1.
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We want greater than or equal to zero, so use the POSITIVE intervals and use open and closed circles to decide [ ] or ( )
[ -3, -1) U (1, ∞)
Write Equation given a sketch Remember:
Cross is odd multiplicity.
Touch is even multiplicity.
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Rational Root Theorem 1) If all coefficients are integers, list all possible combinations. Remember, p are all the factors of the constant, q are all the factors of the leading coefficient.
2) Test each possible zero until you find a factor. A factor HAS A REMAINDER OF ZERO!
3) Repeat the process until you get a quadratic or a factorable equation.
4) Find other zeros by quadratic formula or factoring.
Note: If all coefficients are not integers, you cannot use rational root theorem. In this case, it will most likely be a calculator problem where you will estimate and use those zeros.
Note 2: It’s possible for a zero to repeat, remember that.
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Polynomial Information Degree, end behavior, parent function.
X and Y intercepts
Cross or Touch at x-intercepts.
Determine max\min with calculator.
Draw Graph by Hand
Range
Intervals of increase and decrease
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Parent function y = x3
End behavior follows the parent function y = x3
Remember, parent function matches up with the degree.
x-intercept, y = 0. y-intercept, x = 0
x-intercept 0 = (x-1)2(x+2)
(1,0) (-2,0)
y-intercept y = (0-1)2(0+2)
(0, 2)
Even multiplicity, Touch. Odd multiplicity, Cross
Touch at x = 1
Cross at x = -2
Max (-1, 4)
Min (1, 0)
Note, you can have more than one max and min. Also round to hundredths if necessary.
Max (-1, 4)
Min\x-int (1, 0)
x-int (-1, 0)
y-int (0, 2)
Notice how it touches at x = 1 and crosses at x = -2
Range, lowest y-value to highest y-value. If necessary, look at the y-values of the min and max to help determine range, and use either [ or ] to include that y-value.
(-∞, ∞)
Look at the x-values of the max and mins to help determine intervals. Use parenthesis.
Increase: (- ∞,-1) U (1, ∞)
Decrease: (-1, 1)
Rational Functions\Asymptotes
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1) Domain
2) Reduce Equation
3) Intercepts
4) Even\Odd
5) Holes & Vertical Asymptotes
6) Horizontal or Oblique Asymptotes.
7) Sketch with key points.
Domain: Find where the denominator equals zero before you reduce. It is possible you may have to use quadratic formula or have radicals in the denominator.
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f(x) = f(-x) even
f(-x) = -f(x) odd
Not even
Not odd
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Where the factor cancels out, there is a hole there. To find the y-value of the hole, plug in the x into the reduced equation.
After you reduce, the vertical asymptote is where the denominator equals zero. Remember to put ‘x =‘
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When the degree on bottom is bigger, it is proper, horizontal asymptote is y = 0
When the degree on top is bigger, it is improper, use long division. It will be either horizontal or oblique. REMEMBER PLACEHOLDERS!
Don’t forget ‘y =‘
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Complex Numbers
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The important thing to remember is to put these numbers back into standard form: a + bi
