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TRANSCRIPT
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Graphing Polynomials
Putting it ALL together!
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Let’s graph!
• What do we know??? – Basic shape of even or odd degree function. – Flipped up or down? – End behavior – Roots – # of turning points – Y-Intercept All we need is to find a few points in the middle!!! – To do this, make a table with roots and “fill
in the integer gaps”. • Substitution • Synthetic Division
X Y
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Review of what we’ve learned so far.
• Standard form:
• Degree of polynomial: • Leading coefficient:
• Max # of roots:
• Max # of turning points:
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Degree of the polynomial determines the SHAPE OF THE GRAPH.
Odd Even
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Leading Coefficient tells us how stretched and flipped the graph will be
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Leading Coefficient tells us how stretched and flipped the graph will be
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DEGREE & Leading Coefficient TOGETHER determine end behavior
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Roots tell us….
(You all better know this by now!!!)
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Roots tell us…. Where the graph crosses the x-axis!
# of roots = degree of function
Remember… not all will be REAL.
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What if I get a “double root”
• If you get a double root.. It bounces off the axis rather than going thru. – Just like you already saw with y = x2 and every other
quadratic. – y = x2 – 4x + 4
• If it’s a triple root… it doesn’t bounce.
• See a pattern?
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Y-Intercept
• Been there… done that…
2( ) 3 40f x x x
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The “u-turns” in the middle of the function are caused by the terms in the middle of the polynomial.
• We only know how many there can be at most.
• So… we must use roots and test points to determine how many there actually are.
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TEST POINTS
• You have 2 options
– 1.) Substitution (Ms. Hale’s favorite)
– 2.) Remainder Theorem (Ms. Hale has to teach you this.) using Long or Synthetic Division
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Substitution
• Use the roots and y intercept in table. Substitute integer values in between.
X Y
2( ) 3 40f x x x
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Remainder Theorem
• What makes something a root?
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Remainder Theorem
• What makes something a root?
– It has a y-coord. = 0
– When you divide, you get a remainder of 0.
– So… then for every point that’s not a root, we wouldn’t have a y-coord. = 0 and our remainder wouldn’t = 0.
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What does a remainder tell us?
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What does a remainder tell us?
• How much a divisor is “off” from being a factor.
• So if it’s 0 off it’s a factor (and therefore root), if it’s 15 off, it’s not… its 15 higher than where it needs to be in order to be considered a root.
• … Let’s put it together with a picture.
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Why in the world would division give the same answer as substitution????
• Let’s check it out via desmos.com
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Remainder Theorem • The remainder of long division or synthetic
division by (x – r) = the function value at r.
– Cool huh???
2 3 40
7
x x
x
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Graphs of the polynomials with everything …
X Y
Degree: LC: Flipped? Shape/End Behavior: L: R: Y-int: Max # of U-Turns: Roots: (see hwk)
1.) 3 2( ) 6 8f x x x x
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Work to solve for other points using remainder theorem or substitution.
X Y
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Graphs of the polynomials with everything …
X Y
Degree: LC: Flipped? Shape/End Behavior: L: R: Y-int: Max # of U-Turns: Roots: (see hwk)
1.) 3 2( ) 6 8f x x x x
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Your Assignment • Required for each problem:
1.)
2.)
3.)
4.)
5.)
6.)
7.)
3 2( ) 6 8f x x x x
3 2( ) 2 3f x x x x
3 2( ) 3 4 12f x x x x
3 2( ) 1f x x x x
3 2( ) 2 2f x x x x
3 2( ) 30f x x x
3 2 2( ) 2 14 2f x x x x x
X Y
Degree: LC: Flipped? Shape/End Behavior: Y-int: Max # of U-Turns: Roots: (see hwk)
Graph: Highlighted and labeled.