Graphing Polynomials

Putting it ALL together!

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

Review of what we’ve learned so far.

• Standard form:

• Degree of polynomial: • Leading coefficient:

• Max # of roots:

• Max # of turning points:

Degree of the polynomial determines the SHAPE OF THE GRAPH.

Odd Even

Leading Coefficient tells us how stretched and flipped the graph will be

Leading Coefficient tells us how stretched and flipped the graph will be

DEGREE & Leading Coefficient TOGETHER determine end behavior

Roots tell us….

(You all better know this by now!!!)

Roots tell us…. Where the graph crosses the x-axis!

# of roots = degree of function

Remember… not all will be REAL.

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?

Y-Intercept

• Been there… done that…

2( ) 3 40f x x x

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.

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

Substitution

• Use the roots and y intercept in table. Substitute integer values in between.

X Y

2( ) 3 40f x x x

Remainder Theorem

• What makes something a root?

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.

What does a remainder tell us?

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.

Why in the world would division give the same answer as substitution????

• Let’s check it out via desmos.com

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

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

Work to solve for other points using remainder theorem or substitution.

X Y

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

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.