# 5.6 – Graphing Polynomial Functions

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. $f(x)=x^3-3x-2$
• $f(x)=(x-2)(x+1)(x+1)$
• $f(x)=(x-2)(x+1)^2$
• The y-intercept is: (0,-2)
• The x-intercepts are: (2, 0) and (-1, 0)
• (-1, 0) is a double root
• In the following examples, notice that the graph will touch the x-axis 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 x-values get very small, the function, f(x), gets very small
• $x \rightarrow -\infty$
• $f(x) \rightarrow -\infty$
• As the x-values get very large, the function, f(x), gets very large
• $x \rightarrow \infty$
• $f(x) \rightarrow \infty$

• Y-Intercepts
• When x = 0, you can find the y-intercept
• Notation: (0, c)
• Where a graph crosses the y-axis

• Video: Finding x and y Intercepts and Graphing the Function

• When the multiplicity is odd, the graph crosses the x-axis.
• $(x-4)^3$
• $(x+1)(x-4)^3$
• When the multiplicity is even, the graph touches the x-axis.
• $(x-3)^2$
• $(x+2)(x-3)^2$
• Remember, exponents (powers) represent x-intercepts 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 x-axis one time (even if there are many of the same root).
• Try the practice on Pg 12 before taking the quiz, seriously!

• 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 y-coordinate.
• 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:
• $f(x)=-x^3+x^2-4x+2$
• 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 x-values going forever in the left (negative) and right (positive) directions.
• It doesn’t matter that it is flipped over the x-axis.
• $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$
• $x \rightarrow \infty$, $f(x) \rightarrow \infty$

• Ex 3. What are the x-intercepts of the function:
• $f(x)=x^2-81?$
• The roots ARE the x-intercepts.
• Since this quadratic function can be factored into: (x-9)(x+9), you can solve for the roots: 9 and -9.

• Ex 4. Which of the following is an x-intercept of the function:
• $f(x)=x^3+3x^2-10x-24?$
• Answer Possibilities:  a. -4, b. -3, c. 2, d. 4
• Test each one using substitution.
• $f(-4)=(-4)^3+3(-4)^2-10(-4)-24$
• $f(-4)=-64+3(16)-10(-4)-24$
• $f(-4)=-64+48+40-24$
• $f(-4)=0$
• Which means -4 is a root

• Ex 5. Which polynomial function is graphed below?
• Notice that this is an s-curve, so it is probably a cubic.
• $f(x)=(x+3)^2(x-1)$
• If you expand the factors, you get:
• $f(x)=(x+3)(x+3)(x-1)$
• $f(x)=(x^2+6x+9)(x-1)$
• $f(x)=(x^3+6x^2+9x-x^2-6x-9)$
• $f(x)=x^3+5x^2+3x-9$
• This is a cubic graph with a positive leading coefficient, so it starts low and ends high.

• Ex 6. What is the maximum number of possible extreme values for the function:
• $f(x)=x^3+4x^2-3x-18?$
• There is always 1 less extreme value than there are roots (see the largest exponent).

• 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 x-values are marked by increments of 2, and the y-values are marked by increments of 5.
• Ex 8. How many roots does the polynomial function, $y=(x-8)(x+3)^2$ have?