Polynomial Curve
Sketching
Warm up
When the polynomial 2x2 + bx - 5 is divided by x - 3, the remainder is 7.
(a) Determine the value of b.
Warm up

 Solve by factoring the polynomial completely.


  (a) x 3 + 5x 2 + 2x - 8 = 0
Use your TI 83 to find the roots of the
polynomial.
        x3 - 2x2 - 5x + 6


Answer x = -2, 1, 3
Degree n of a polynomial is odd

        The function has opposite behaviour


If the leading coefficient is >0

        T...
When the degree n of a polynomial is even,
then the graph has similar behaviour on the
left as on the right


If the leadi...
Graphing Polynomial Functions


Appearance      ƒ(x) = x n


             Where n is even, the     Where n is odd, the
   ...
Graphing Polynomial Functions
 Roots
     The maximum number of roots for any polynomial function is
     equal to the deg...
Graphing Polynomial Functions

 Sketching


  Step 1: Find the y-intercept (let x = 0)


  Step 2: Find all roots. (Use ra...
Factor the polynomial completely. Sketch the graph.
    ƒ(x) = x 3+ 5x 2 + 2x - 8
Sketch the graph of
Jan. 13 polynonial sketching
Jan. 13 polynonial sketching
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Jan. 13 polynonial sketching

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Jan. 13 polynonial sketching

  1. 1. Polynomial Curve Sketching
  2. 2. Warm up When the polynomial 2x2 + bx - 5 is divided by x - 3, the remainder is 7. (a) Determine the value of b.
  3. 3. Warm up Solve by factoring the polynomial completely. (a) x 3 + 5x 2 + 2x - 8 = 0
  4. 4. Use your TI 83 to find the roots of the polynomial. x3 - 2x2 - 5x + 6 Answer x = -2, 1, 3
  5. 5. Degree n of a polynomial is odd The function has opposite behaviour If the leading coefficient is >0 The graph rises to the right and falls to the left 3 2 x +2x +1 If the leading coefficient is <0 The graph rises to the left and falls to the right 3 2 -2x +2x +1
  6. 6. When the degree n of a polynomial is even, then the graph has similar behaviour on the left as on the right If the leading coefficient >0 the graph rises on the left and rises on the right 4 3 2 x + x - 2x + x + 1 If the leading coefficient <0 the graph falls on the left and falls on the right -x4 - 2x3 + 2x2 + x + 1
  7. 7. Graphing Polynomial Functions Appearance ƒ(x) = x n Where n is even, the Where n is odd, the graph looks like this: graph looks like this:
  8. 8. Graphing Polynomial Functions Roots The maximum number of roots for any polynomial function is equal to the degree of the function. Examples: Cubic Quartic Quintic ƒ(x) = x 3 ƒ(x) = x 4 ƒ(x) = x 5 max. # of roots 3 4 5
  9. 9. Graphing Polynomial Functions Sketching Step 1: Find the y-intercept (let x = 0) Step 2: Find all roots. (Use rational roots theorem if necessary.) Step 3: Determine the sign of the function over the intervals defined by the roots. Step 4: Sketch the graph
  10. 10. Factor the polynomial completely. Sketch the graph. ƒ(x) = x 3+ 5x 2 + 2x - 8
  11. 11. Sketch the graph of

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