5.8-Graphs of Polynomials

• 1. Plot x-intercepts (solutions: opposites of factors)• 2. Decide if graph touches or goes through at each

zero• 3. Determine LEFT & RIGHT side behaviors and draw

graph from right and left most zeros• 4. Plot y-intercept (put 0 in for all x’s)• 5. Fill in graph with appropriate number of turns (2

humps disappear with each PAIR of imaginary zeros) May use calculator to find local max and min values (or plot points between x-intercepts)

Behavior at repeated zeros

• Repeated zeros happen if factor (and zero) exist more than once (power with factor)

• ODD # of times: graph goes THROUGH axis• EVEN # of times: Graph only TOUCHES axis and

goes back the same way (A turning point or local/relative maximum or minimum)

• Local(relative) maximum: high point of a hump (plot points or use calculator)

• Local(relative) minimum: low point of a hump

End Behaviors of graph

• RIGHT SIDE: – Leading coefficient

• Positive = up on right side• Negative = down on right side

• LEFT SIDE:– Degree (Largest power)

• Even = same direction as right side• Odd = opposite direction as right side

• TURNS (humps):– Degree (Largest power) subtract 1

Examples: Graph

• 1. f(x)= (x – 2)²(x + 1)

Examples: Graph

• 2. -2(x – 1)(X +4)³(x – 3)²

Examples: Graph

• 3. f(x)= 0.25(x+2)(x-1)(x-3)

• 4. f(x)= 2(x-1)²(x-4)

Examples:

• 5. # 15 p. 390

• 6. # 17 p. 390

• 7. # 19 p. 390