By: Deanna Carbone, Jacqueline DiSalvatore, and Alyssa Fanelli Graphing Polynomials By: Deanna Carbone, Jacqueline DiSalvatore, and Alyssa Fanelli

Definitions Real Zeros – the value of x when y=0; the y-intercepts Rational Roots Theorem – p/q is a rational zero of f, a polynomial of degree 1 or higher or the coefficient is an integer Multiplicity – when there are multiple of the same real zeros [ex: (x+2)2] Sign Analysis – testing the intervals for values between the real zeros Maximum – highest y-value and turning point of the graph Minimum – lowest y-value and turning point of the graph Y-intercept – point on the graph when y=0 Intervals of increasing and decreasing – where y increases or decreases on the graph; determined by the change in y, but written using x-values Degree of a polynomial – highest exponent in the equation

Step 1 Find the real zeros Factor the equation completely (use Rational Roots Theorem if needed) Look for multiplicities Even multiplicity – graph touches the x-axis Odd multiplicity – graph goes through the x-axis

Step 2 Test intervals Choose values between the real zeros found from Step 1 Plug these values into each factor of the original polynomial and determine whether the answer is positive or negative Once all values are tested, multiply the signs of each answer to determine the end behavior The end behavior should relate to the degree of the polynomial Even degree – same end behavior Odd degree – opposite end behavior

Step 3 Determine the number of turning points Subtract 1 from the degree of the polynomial to determine the number of maximums and minimums

Step 4 Graph the polynomial Find the y-intercept by plugging 0 into the original equation Graph Analyze the graph X-intercepts Maximums and minimums (from calculator) Intervals of increasing and decreasing

Odd Degree P(x) = x3+8x2+11x-20 = (x+4)(x+5)(x-1) Step 1 Find real zeros: x= -4, x= -5, x=1 No multiplicities

Odd Degree (cont’d) Step 2 Test intervals -6 -4.5 0 2 x+4 - - + + -6 -4.5 0 2 x+4 - - + + x+5 - + + + x-1 - - - + - + - +

Odd Degree (cont’d) Step 3 Step 4 Find the turning points (Degree – 1) Graph (on next slide) Y-intercept = -20 X-intercepts = -4, -5, 1 Max: (-4.5, 1.4) Min: (-.81, -24.2) Intervals of increasing: (-∞, -4.5] [-.81, ∞) Interval of decreasing: [-4.5, -.81]

Odd Degree (cont’d)

Even Degree P(x) = x4-2x3-3x2 = x2(x2-2x-3) Step 1 Find real zeros: x=0, x=3, x= -1 0 has a multiplicity of 2

Even Degree (cont’d) Step 2 Test intervals -2 -1/2 1 4 x2 + + + + x-3 - - - + x+1 - + + + + - - +

Even Degree (cont’d) Step 3 Step 4 Find the turning points (Degree-1) Graph (on next slide) Y-intercept = 0 X-intercepts = 0, 3, -1 Max: (0,0) Mins: (-.68, -.544) (2.18, -12.4) Intervals of increasing: [-.686, 0] [2.18, ∞) Intervals of decreasing: (-∞, -.686] [0, 2.18]

Even Degree (cont’d)

Even Degree with a Multiplicity P(x) = x4+x3-12x2 = x2(x+4)(x-3) Step 1 Find real zeros: x=0, x= -4, x=3 0 has a multiplicity of 2

Even Degree with a Multiplicity (cont’d) Step 2 Test intervals -5 -1 1 5 x2 + + + + x+4 - + + + x-3 - - - + + - - +

Even Degree with a Multiplicity (cont’d) Step 3 Find the turning points (Degree-1) (4-1) = 3 turning points Step 4 Graph (on next slide) Y-intercept = 0 X-intercepts = -4, 3, 0 Max: (0,0) Mins: (2.10, -24.21) (-2.85, -54.64) Intervals of increasing: [-2.85, 0], [2.10, ∞) Intervals of decreasing: (-∞, -2.85], [0, 2.10]

Even Degree with a Multiplicity (cont’d)