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Warm-Up

Warm-Up. Write the following polynomial in its factored form and then sketch it: Double root at (-5,0), triple root at (-3,0), single root at (0,0) and a triple root at (4,0). The leading term is positive. Expand the following terms:. Exact Graphs of Polynomials. Learning Targets.

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Warm-Up

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  1. Warm-Up • Write the following polynomial in its factored form and then sketch it: • Double root at (-5,0), triple root at (-3,0), single root at (0,0) and a triple root at (4,0). The leading term is positive. • Expand the following terms:

  2. Exact Graphs of Polynomials

  3. Learning Targets • Look at how “a” changes the equation • Determine the equation of a polynomial • Writing the polynomial in standard form

  4. Recap • Up to this point • We have only looked at the roots and degrees of a polynomial • This information allowed us to sketch our graphs • However, we have not been able to determine the exact equations

  5. What does “” do… • The “” term is the coefficient of the leading term • The value for “” determines the vertical stretch for our graph

  6. Vertical Stretch Factor • We have seen this factor come into play before… where? • When transforming functions we had to solve for this term by looking a non-linear functions with an included point • This factor would determine whether our graph was stretched or compressed

  7. “” in polynomials • With polynomials the “” factor is no different • This is going to determine what happens to the graph between each root

  8. Write out the equation for this graph and graph it on your calculator: What is the “” term?

  9. How to find “” • What information did we need in order to find the vertical stretch/compression factor in Ch.4? • In order to find “” we have to know a point on the line. • This allows us to evaluate the function at a certain point and isolate the “” term

  10. Write out the equation for this graph and graph it on your calculator: Now find “”

  11. Solving • Use point to solve for Now Graph it on your calculator

  12. Effect of Polynomials • Polynomials tend to produce high values for outputs • This is because of the repeated exponential increase that occurs within each term • Experiment with the following function by using different values for “”

  13. What happens with different “” values • This stretch/compression term has major implications on how our polynomial will behave

  14. Determine the “” values

  15. Determine the “” values

  16. Determine the “” values

  17. Putting it all together – graphically… • The last piece to deciding the exact equation for polynomials based on their graphs is finding the value for “” • In order to find the “” value we must be given a point on the line

  18. Finding “” with only an equation • Most of the time we will be given only an equation and asked to graph it • There are two versions of an equation that we will explore: • Factored Form (FF) • Standard Form (SF)

  19. Factored/Standard Form • In both forms of a polynomial equation our “” value will be given to us • It is always the first coefficient in our equation • FF: • SF:

  20. Practice: Going from FF to SF • Our ultimate goal is transform our polynomials from SF to FF so we can find solutions and graph them • However, we must be comfortable with going from FF to SF first • To do this we must distribute each term in its factored form and simplify the end result

  21. Example:

  22. Practice • Put the following polynomials into their SF

  23. For Tonight: • Worksheet

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