Key Terms
 Concavity – A description of the direction of a graph’s curve.
 Inflection Point – A point at which a graph’s concavity changes.
Review
 End Behavior of a Graph
 How a graph starts and ends
 See Algebra 2B, Section 5.2
 Large and Small Numbers
 Numbers get “smaller” as you move to the left on the number line.
 Numbers get “larger” as you move to the right on the number line.
 Numbers close to zero are not necessarily small.
 Ex. Zero is smaller than 4, and 10 is smaller than zero.
 Ex. Zero is larger than 10, and 4 is larger than zero.
 Ex. 430 > 435,351,010
 Ex. 500 < 0.00000000000000000000001
 Multiplicity – When one factor appears more than once
 Ex.
 The yintercept is: (0,2)
 The xintercepts are: (2, 0) and (1, 0)
 (1, 0) is a double root
 In the following examples, notice that the graph will touch the xaxis at an extreme or at a curve
 Maximums and Minimums
 Relative maxima are sections of the graph that look like hills, but are not the highest point(s) on the graph.
 Relative minima are sections of the graph that look like valleys, but are not the lowest point(s) on the graph.
 Absolute maximum – the highest point on the graph (the tallest hill)
 Absolute minimum – the lowest point on the graph (the lowest valley)
Notes
 End Behaviors
 If the degree of a function is even, the end behaviors are the same (both up or both down).
 If the degree of a function is odd, the end behaviors are different (one is up and one is down).
 Graph Patterns
 Positive Leading Coefficient & Even Power: Both Ends UP
 Negative Leading Coefficient & Even Power: Both Ends DOWN
 Positive Leading Coefficient & Odd Power: Left DOWN, Right UP
 Negative Leading Coefficient & Odd Power: Left UP, Right DOWN
 Notation
 As the xvalues get very small, the function, f(x), gets very small
 As the xvalues get very large, the function, f(x), gets very large
 As the xvalues get very small, the function, f(x), gets very small
 YIntercepts
 When x = 0, you can find the yintercept
 Notation: (0, c)
 Where a graph crosses the yaxis
 Video: Finding x and y Intercepts and Graphing the Function
 More About Multiplicity
 When the multiplicity is odd, the graph crosses the xaxis.
 When the multiplicity is even, the graph touches the xaxis.
 Remember, exponents (powers) represent xintercepts and changes in direction.
 Multiplicity is impossible to see in a graph, as it doesn’t change direction, and it only crosses or touches the xaxis one time (even if there are many of the same root).
 Try the practice on Pg 12 before taking the quiz, seriously!
 When the multiplicity is odd, the graph crosses the xaxis.
 Extreme Values
 Extreme values are valleys and hills seen in graphs.
 If a polynomial function has degree n, the graph will have at most n − 1 relative maximum or relative minimum values.
 Ex. if a polynomial function has a degree of 4, its graph has at most 4 − 1, which is equal to 3, extreme values. It may have 1, 2, or 3 extreme values. It cannot have more than 3, but it can have less.
 Graphs that continue up or down for infinity do NOT have any extreme values
 They may relative maximums and minimums, but not absolute (or extreme).
 Extremes are denoted by the ycoordinate.
 Ex. If a relative maximum has its highest point at (3, 4), the relative maximum is 4 while the x = 3.
 Ex. if a relative minimum has its lowest point at (3, 4), the relative minimum is 4 while x = 3.
 Inflection Point
 Graphing Completely
Examples
 Ex 1. Which of the following correctly describes the end behavior of the polynomial function:
 Answer: Left end up, right end down
 Ex 2. Which of the following notations correctly describe the end behavior of the polynomial graphed below?
 Domain: notice that the graph is a parabola, with xvalues going forever in the left (negative) and right (positive) directions.
 It doesn’t matter that it is flipped over the xaxis.
 Answer:
 ,
 ,
 Ex 3. What are the xintercepts of the function:

 The roots ARE the xintercepts.
 Since this quadratic function can be factored into: (x9)(x+9), you can solve for the roots: 9 and 9.

 Ex 4. Which of the following is an xintercept of the function:
 Answer Possibilities: a. 4, b. 3, c. 2, d. 4
 Test each one using substitution.
 Correct answer: 4

 Which means 4 is a root

 Answer Possibilities: a. 4, b. 3, c. 2, d. 4
 Ex 5. Which polynomial function is graphed below?
 Notice that this is an scurve, so it is probably a cubic.
 Answer:
 If you expand the factors, you get:

 This is a cubic graph with a positive leading coefficient, so it starts low and ends high.

 If you expand the factors, you get:
 Ex 6. What is the maximum number of possible extreme values for the function:

 There is always 1 less extreme value than there are roots (see the largest exponent).
 Answer: 2

 Ex 7. Which point is the best approximation of the relative maximum of the polynomial function graphed below?
 Look for the point where there is a hill. What is the coordinate (x, y) of that hill’s top?
 Also notice that the xvalues are marked by increments of 2, and the yvalues are marked by increments of 5.
 Answer: (–3.6, 17)
 Ex 8. How many roots does the polynomial function, have?
 Since 2 of the xvalues are the same, they would have the same exact root (a double root).
 So, the roots are: 8 and 3.
 Answer: There are 2 unique roots.