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Math 1314 Lesson 12 Curve Analysis (Polynomials)
This lesson will cover analyzing polynomial functions using GeoGebra.
Suppose your company embarked on a new marketing campaign and was able to track sales based on it. The graph below gives the number of sales in thousands shown t days after the campaign began.
Now suppose you are assigned to analyze this information. We can use calculus to answer the following questions:
When are sales increasing or decreasing? (Note that the graph stops at t = 40.) What is the maximum number of sales in the given time period? Where does the growth rate change? Etc. Calculus can’t answer the “why” questions, but it can give you some information you need to start that inquiry.
There will be several features of a polynomial function that we’ll need to find. Let’s start with a few College Algebra topics.
An example of a polynomial function is f ( ) x = ( x − 2)( x −1) ( 3 x + 1)^2. Its graph looks like:
*The domain of any polynomial function is ( −∞ ∞, ). Polynomial functions have no asymptotes.
*To find the roots/zeros/ x- intercepts/solutions of any function, set the function equal to zero and solve for x. Or you may simply use the root or roots command in GGB. *To find the y- intercept for any function, set x = 0 and calculate. *The range of any polynomial is easier found by looking at the graph of the function.
Example 1: Let f ( ) x = x^3^ − 3 x^2 − 13 x + 15. Enter the function in GGB.
a. Find any x -intercepts of the function. Command: Answer:
b. Find any y- intercept of the function. Command: Answer:
Intervals on Which a Function is Increasing/Decreasing
Definition : A function is increasing on an interval ( a, b ) if, for any two numbers and in
( a, b ), , whenever. A function is decreasing on an interval ( a, b ) if, for
any two numbers and in ( a, b ), , whenever.
In other words, if the y values are getting bigger as we move from left to right across the graph of the function, the function is increasing. If they are getting smaller, then the function is decreasing. We will state intervals of increase/decrease using interval notation. The interval notation will consists of corresponding x- values wherever y- values are getting bigger/smaller.
Example 2: Given the following graph of a function, state the intervals on which the function is: a. increasing. b. decreasing.
We can use calculus to determine intervals of increase and intervals of decrease. A function can change from increasing to decreasing or from decreasing to increasing at its critical numbers , so we start with a definition of critical numbers:
The critical numbers of a polynomial function are all values of x that are in the domain of f
where f ′ ( x ) = 0 (the tangent line to the curve is horizontal).
A function is increasing on an interval if the first derivative of the function is positive for every number in the interval. A function is decreasing on an interval if the first derivative of the function is negative for every number in the interval.
x 1 x 2 f ( x 1 )< f ( x 2 ) x 1 (^) < x 2 x 1 x 2 f ( x 1 )> f ( x 2 ) x 1 (^) < x 2
To find the intervals on which a given polynomial function is increasing/decreasing using GGB:
- Use GGB to graph the derivative of the function.
2. Find any critical numbers. ( Recall that the critical numbers occur whenever f ′^ ( x ) = 0 ;
hence, simply find the zeros of f ′ .)
- Create a number line, subdividing the line using the critical numbers.
- Use the graph of the derivative (or compute the value of a test point) to determine the sign (positive or negative) of the y values of the derivative in each interval and record this on your number line.
- In each interval in which the derivative is positive, the function is increasing. In each interval in which the derivative is negative, the function is decreasing.
Example 5: Let f ( ) x = x^5^ − 16 x^2 + 4 x. Using GGB find: a. any critical numbers. Command: Answer:
b. any intervals where the function is increasing/decreasing.
Relative Extrema
The relative extrema are the high points and the low points of a function. A relative maximum is higher than all of the points near it; a relative minimum is lower than all of the points near it.
A relative maximum or a relative minimum can only occur at a critical number.
You can use the same number line that you created to determine intervals of increase/decrease to find the x coordinate of any relative extrema. Use these three statements to determine if a critical number generates a relative extremum. Once you find that x = c generates a relative extremum, you can find the y coordinate of the relative extremum by computing f c ( ) or for a more
accurate answer use the extremum command in GGB.
- If the sign of the derivative changes from positive to negative at a critical number, x = c , then
the function has a relative maximum at the point ( c , f ( c )).
- If the sign of the derivative changes from negative to positive at a critical number, x = c , then
the function has a relative minimum at the point ( c , f ( c )).
- If the sign of the derivative does not change sign at a critical number, x = c , then the function
has neither a relative maximum nor a relative minimum at the point ( c , f ( c )).
Example 6: Find any relative maximum and relative minimum for f ( ) x = x^5^ − x^4.
Example 7: Find any relative extrema: f ( ) x = 0.25 x^6 + x + 4
Command: Answer:
Example 9: State intervals on which the function is concave upward and intervals on which the
function is concave downward: 5 2
f x = x − x − x −
Command:
You’ll also need to be able to identify the point(s) where concavity changes. A point where concavity changes is called a point of inflection.
You can use the same number line that you created to determine concavity intervals to find the x coordinate of any inflection points. Use the following two statements to determine if a zero of the second derivative generates an inflection point.
- If the sign of the second derivative changes from positive to negative or from negative to
positive at a number, x = c , then the function has an inflection point at the point ( c , f ( ) c ).
- If the sign of the second derivative does not change sign at a number, x = c , then the function
does not have an inflection point at the point ( c , f ( ) c ).
Once you find that x = c generates an inflection point, you can find the y coordinate of the inflection point by computing f c ( ).
Example 10: Given 5 2
f x = x − x − x − , find any inflection points.
Example 11: Given , find any inflection points. Command: Answer:
f ( x )= x^3 − 3 x^2 − 24 x + 32
Analyzing a Function
Example 12: The graph given below is the graph of a polynomial function f. Which of the statements below is/are true?
- The function has a horizontal asymptote.
- The function is only increasing.
- The function has one relative maximum and no relative minimum.
- The function has two points of inflection.
Example 13: Analyze the function:
a. Domain
b. Coordinates of any zeros. Command: Answer:
f x = x − x + x +
