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Characteristics of Polynomial Functions2.1
POLYNOMIAL FUNCTIONS
2.1 Characteristics of Polynomial Functions p. 85
2.2 The Remainder Theorem p. 97
2.3 The Factor Theorem p. 109
REVIEW PRACTICE SECTION p. 131
2.4 Analyzing Graphs of Polynomial Functions p. 121
Determine an equation for each of the following functions:
Youโve already studied Polynomial functions! Remember?
In Math 10C you studied Linear Functions And in Math 20-1 you studied Quadratic Functions
๐๐ = ๐๐๐๐+ ๐๐
We saw how equations of linear functions can be written in the form
๐๐ = ๐๐(๐๐โ ๐๐)๐๐+๐๐
โฆ And equations of quadratic functions can be written
Where ๐๐ is the slope of the line, and ๐๐ is the ๐๐-intercept
Where ๐๐ is the vertical stretch, and the coordinates of the vertex are (๐๐,๐๐).
(โ,๐๐)
(๐๐, 0) (๐๐, 0)
๐๐ = ๐๐(๐๐โ๐๐)(๐๐โ ๐๐)Note that the quadratic functions can also be written in the form
Where ๐๐, ๐๐ are ๐๐-intercepts
๐๐ = ๐๐(๐๐โ ๐๐)Note that the linear functions can also be written in the form
Where ๐๐ is the ๐๐-intercept
(0,๐๐)(๐๐, 0)
1
(1, 0)
(โ1,โ2)
(โ3, 0)
2
Hint:The vertical stretch here, ๐๐, is ๐๐/๐๐
These are degree 1 Polynomial Functions These are degree 2 Polynomial Functions
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2.1 Characteristics of Polynomial Functions
Page|86
Identifying Polynomial FunctionsClass Example 2.11
(a) ๐ฆ๐ฆ = 4๐ฅ๐ฅ5 โ 3๐ฅ๐ฅ2
Identify which of the following are polynomial functions. For each that is a polynomial function; state the degree and leading coefficient:
(b) ๐ฆ๐ฆ = 2 ๐ฅ๐ฅ + 5๐ฅ๐ฅ (c) ๐ฆ๐ฆ = 3 ๐ฅ๐ฅ + 7 (d) ๐ฆ๐ฆ = โ2๐ฅ๐ฅ3 + 5๐ฅ๐ฅโ1
Worked Example
Identify which of the following are polynomial functions. For each that is a polynomial function; state the degree and leading coefficient:
(a) ๐ฆ๐ฆ = โ6๐ฅ๐ฅ2 +3๐ฅ๐ฅโ 8 (b) ๐ฆ๐ฆ = 3๐ฅ๐ฅ4 โ 2๐ฅ๐ฅ5 โ 3๐ฅ๐ฅ2 โ 1 (c) ๐ฆ๐ฆ = 5๐ฅ๐ฅ2 โ 33 ๐ฅ๐ฅ + 1
Solution: (a) The middle term can be written 3๐ฅ๐ฅโ1, which is NOT POLYNOMIAL as exponent of ๐ฅ๐ฅ is not a whole number
(b) All exponents are whole numbers, and all coefficients are real numbers. Hello, you POLYNOMIAL FUNCTION.Degree is 5 (degree of entire poly function is that of highest degree term). Leading coefficient is โ๐๐. (the terms are not in descending order of degree โ the 2nd term should be re-arranged to the โfrontโ!)
(c) The middle term can be written 3๐ฅ๐ฅ1/3, which is NOT POLYNOMIAL as the exponent is not a whole number
A polynomial function, such as a linear function, quadratic, or cubic, involves only non-negative integer powers of ๐ฅ๐ฅ.
Any polynomial function can be written in the form๐๐ ๐ฅ๐ฅ = ๐๐๐๐๐ฅ๐ฅ๐๐ + ๐๐๐๐โ1๐ฅ๐ฅ๐๐โ1 + ๐๐๐๐โ2๐ฅ๐ฅ๐๐โ2 + โฆ + ๐๐2๐ฅ๐ฅ2 + ๐๐1๐ฅ๐ฅ1 + ๐๐0
Where: ๐๐ is a whole number, representing the degree of the function๐๐๐๐ to ๐๐0 are real numbers, representing the coefficients, with ๐๐๐๐designated the leading coefficient (coeff. of the highest degree term)
This general formula may look complicated, but a few polynomial function examples should show its simplicity:
๐ฆ๐ฆ = 8 This is degree 0 (constant function)
๐ฆ๐ฆ = โ5๐ฅ๐ฅ + 4 This is degree 1 (linear), with a leading coefficient of โ5
๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 3 This is degree 2 (quadratic), with a leading coefficient of 2
๐ฆ๐ฆ = ๐ฅ๐ฅ3 +12๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ โ 1 This is degree 3 (quadratic), with a leading coefficient of 1
๐๐ ๐ฅ๐ฅ = 3๐ฅ๐ฅ4 โ 2๐ฅ๐ฅ3 + 5๐ฅ๐ฅ โ 8 This is degree 4 (quartic), with a leading coefficient of 3
๐๐ ๐ฅ๐ฅ = โ2๐ฅ๐ฅ5 + ๐ฅ๐ฅ4 โ 3๐ฅ๐ฅ2 โ 6 This is degree 5 (quintic), with a leading coefficient of โ2
Polynomial functions can be of any whole number degree ๐๐ โ but for this course weโll only deal with functions where ๐๐ โค 5.
And while the coefficients can be any real number โ weโll mainly stick with integer coefficients.
These examples are all written in descending order of degree, where terms are arranged starting with the highest degree term, starting with the leading coefficient. (The coefficient of the highest degree term)
Defining Polynomial Functions
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Unit 2 โ Polynomial Functions
For both terms, exponents of ๐ฅ๐ฅ are whole numbers. POLYNOMIAL FUNCTION , degree 5 Leading Coeff. 4(a)
Answers from previous page
2.11The first term could be written as 2๐ฅ๐ฅ1/2, which is NOT POLYNOMIAL as the exponent of ๐ฅ๐ฅ is not a whole number. (b)
Thatโs a POLYNOMIAL FUNCTION (Donโt be thrown off by the irrational 3 coefficient โ thatโs allowed!) Polynomial functions must have whole # variable exponents, coefficients can be any real #. degree 1 Leading Coeff. ๐๐
(c)
The last term is NOT POLYNOMIAL as the exponent of ๐ฅ๐ฅ is not a whole number. (d)
A polynomial function can be of any whole number degree, including zero!The graph on the right is of the constant function ๐๐ ๐ฅ๐ฅ = 3, which is a polynomial function of degree zero. Letโs now acquaint ourselves with some examples of polynomial functions, of degree 1 through 5:
๐๐ ๐ฅ๐ฅ = 2๐ฅ๐ฅ โ 3
Degree 1 Linear
Domain: {๐ฅ๐ฅ โ โ}Range: {๐ฆ๐ฆ โ โ}
End Behavior: starts negative in quad III, ends positive in quad I# of intercepts: ๐๐
๐๐ ๐ฅ๐ฅ = โ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ + 3
Degree 2 Quadratic
Domain: {๐ฅ๐ฅ โ โ}Range: {๐ฆ๐ฆ โค 4, ๐ฆ๐ฆ โ โ}
End Behavior: starts negative in quad III, ends negative in quad IV# of intercepts: ๐๐
๐๐ ๐ฅ๐ฅ = ๐ฅ๐ฅ4 โ 9๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 12
Degree 4 Quartic
Domain: {๐ฅ๐ฅ โ โ}Range: {๐ฆ๐ฆ โฅ โ16.9, ๐ฆ๐ฆ โ โ}
End Behavior: starts positive in quad II, ends positive in quad I# of intercepts: ๐๐
๐๐ ๐ฅ๐ฅ = ๐ฅ๐ฅ3 โ 3๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 3
Degree 3 Cubic
Domain: {๐ฅ๐ฅ โ โ}Range: {๐ฆ๐ฆ โ โ}
End Behavior: starts negative in quad III, ends positive in quad I# of intercepts: ๐๐
The functions on the left are odd degree โ and the graphs start and end in the opposite direction. For example, the degree 5 function starts positive and ends negative.
Odd functions have no max or min point, must have at least one ๐ฅ๐ฅ-intercept, and have a range {๐๐ โ โ}.
The functions above are even degree. As such, the graphs start and end in the same direction. For example, the degree 4 function starts positive and ends positive.
Even functions have either a maximum or minimum point, and the range is restricted accordingly.
If the sign of the leading coefficient is positive (the degree 1, 3, and 4 examples above), the graph โends positiveโ, or heading upward, in quad I. And If the degree is even, the graph will have a minimum point.
I wish my lead coeff. wasnโt so negative
Ends positive
If the sign is negative (see the degree 2 and 5 examples), the graph โend negativeโ, or downward, in quad IV. And if its even degree (as with example 2), the graph will have a maximum point.
Classifying Polynomial Functions by Degree๐๐ ๐ฅ๐ฅ = 3
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โ ๐ฅ๐ฅ = โ๐ฅ๐ฅ5 + 4๐ฅ๐ฅ4 + ๐ฅ๐ฅ3 โ 16๐ฅ๐ฅ2 + 12๐ฅ๐ฅ
Degree 5 Quintic
Domain: {๐ฅ๐ฅ โ โ}Range: {๐ฆ๐ฆ โ โ}
End Behavior: starts positive in quad II, ends negative in quad IV# of intercepts: ๐๐
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2.1 Characteristics of Polynomial Functions
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There is a relationship between the degree of a polynomial function and the number of ๐๐-intercepts on the graph.
For a polynomial function of degree ๐๐; the maximum number of ๐ฅ๐ฅ-intercepts is ๐๐.
For example, this degree 4function has no ๐ฅ๐ฅ-intercepts
For example, consider a degree 5 functionSince it starts / ends in the opposite direction (in this case starts positive in quad II, ands negative in quad IV)โฆthere must be at least one ๐๐-intercept
For even degree functions, there can be ๐๐ to ๐๐ ๐ฅ๐ฅ-intercepts.
For odd degree functions, there can be ๐๐ to ๐๐ ๐ฅ๐ฅ-intercepts.
Now, a degree 4 function can also have 1 ๐๐-intercept โฆ or 2 ๐๐-intercepts โฆ or 3 ๐๐-intercepts To a maximum of 4
Note (unlike the functions above), this functionmust have a negative leading coefficient. (ends down)
And we should know how to spot a Polynomial Function Graph!
On the previous page we saw the relationship between the degree of a polynomial function and certain characteristics of the graph. You might next ask โhow can we immediately tell that a graph is of a polynomial function, and not some other function we study in Math 30?
And once again โ great question! I respect your inquisitive nature. Letโs dive into that, with a couple of key distinguishing points:
Have no start or end points, like, for example, radical function graphs.
Polynomial Function
Radical Function
(Domain is Restricted)
Graph starts at this point
Have no vertical asymptotes or any other type of discontinuity, as with rational function graphs.
Graph can be drawn without lifting your pencil.
The first key point is that all polynomial functions have a domain {๐ฅ๐ฅ โ โ}. That means graphs of polynomial functions:
๐๐ ๐ฅ๐ฅ = 0.5 ๐ฅ๐ฅ + 3 โ 2 ๐๐ ๐ฅ๐ฅ =1
(๐ฅ๐ฅ + 2)(๐ฅ๐ฅ โ 3)
The second point is polynomial function graphs have no horizontal asymptotes (like exponential functions) and there is no periodic pattern (as with some trig graphs).
So graphs will always both start and end in either an upward or downward position.
Rational Function
Exponential Function
โ ๐ฅ๐ฅ = (1.5)๐ฅ๐ฅโ3
ENDS upward (pos lead coeff.)
STARTS pointing downward
For example, this graphโฆ.
Graph has vertical asymptotes (again, domain is restricted)
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Page|89
Unit 2 โ Polynomial Functions
Characteristics of Polynomial FunctionsClass Example 2.12
(a) ๐๐ = ๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐
For each of the following polynomial functions, without using your graphing calculator, state:
(b) ๐๐ = ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐๐๐ โ ๐๐๐๐๐๐๐๐ โ ๐๐๐๐
i - The start and end behavior of the graph ii - The number of possible ๐ฅ๐ฅ-interceptsiii - Whether or not the graph will have a minimum or maximum pointiv - The domain of the function and the ๐ฆ๐ฆ-intercept
Use your graphing calculator to determine:v - The ๐ฅ๐ฅ-intercepts of the graph vi - The range of the function
Worked Example
For the polynomial function ๐๐ ๐ฅ๐ฅ = โ๐ฅ๐ฅ4 + 3๐ฅ๐ฅ3 + 7๐ฅ๐ฅ2 โ 15๐ฅ๐ฅ โ 18 ;
Sol.:
Without using your graphing calculator, state:i - The start and end behavior of the graph ii - The number of possible ๐ฅ๐ฅ-interceptsiii - Whether or not the graph will have a minimum or maximum pointiv - The domain of the function and the ๐ฆ๐ฆ-intercept
Use your graphing calculator to determine:v - The ๐ฅ๐ฅ-intercepts of the graph vi - The range of the function
i -
The degree of the function is 4; since itโs even, the graph will start and end in the same direction. And since the leading coefficient is negative (that is, โ-1โ), the graph will end negative / heading downward. Soโฆ. The graph starts negative in quadrant III, and ends negative in quadrant IV.
ii - Degree 4 (even), so there can be between 0 and 4 ๐๐-intercepts. iii - Even degree, graph starts / ends in the same direction. Negative leading coefficient, so which means the
graph ends negative. Therefore the graph will have a maximum point, which can be found graphically.iv - All polynomial functions have domain {๐๐ โ โ}. The ๐ฆ๐ฆ-intercept is the same as the constant value, so (๐๐,โ๐๐๐๐)
So, ๐๐-intercepts are (โ๐๐,๐๐), (โ๐๐,๐๐), and (๐๐,๐๐)
vi -For the range, find the MAXIMUM, which is also in the CALC menu.
v - ๐ฅ๐ฅ-intercepts are the same as the zeros of the function. The zero function is in CALC menu, found be entering
Find the zeros one at a time...
๐๐ = โ๐๐
๐๐ = โ๐๐
Note: sometimes the calc adds decimals. Here, the actual value is just ๐๐.
The range is: {๐๐ โค ๐๐.๐๐๐๐,๐๐ โ โ}Note that the maximum is provided as an approximate value, to the nearest hundredth.
i - Start / end
ii - # of ๐ฅ๐ฅ-ints
iii - Max or min?
iv - Domain:๐ฆ๐ฆ-intercept:
v - Coords of ๐ฅ๐ฅ-ints:
vi - Range:
i - Start / end
ii - # of ๐ฅ๐ฅ-ints
iii - Max or min?
iv - Domain:๐ฆ๐ฆ-intercept:
v - Coords of ๐ฅ๐ฅ-ints:
vi - Range:
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2.1 Characteristics of Polynomial Functions
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More Characteristics of Polynomial FunctionsClass Example 2.13
For the function ๐๐ ๐๐ = โ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐๐๐โ ๐๐๐๐, without using your graphing calculator, state:
i - The start and end behavior of the graph
ii - The number of possible ๐ฅ๐ฅ-intercepts
iii - Whether or not the graph will have a minimum or maximum point
iv - The domain of the function and the ๐ฆ๐ฆ-intercept
Then, use your graphing calculator to determine:v - The ๐ฅ๐ฅ-intercepts of the graph
vi - The range of the function nearest hundredth
Label any intercepts and max / min pointsLastly, sketch the graph on the grid below.
i โ Odd degree, Positive Leading Coefficient โ So starts negative in quadrant III and ends positive in quad I.(a)2.12ii โ Degree 3 (odd), so between 1 and 3 ๐ฅ๐ฅ-intercepts. iii โ Odd degree so no max / miniv โ Domain is {๐๐ โ โ} ๐ฆ๐ฆ-intercept is: ๐ฆ๐ฆ = 0 3 โ 2 0 2 โ โ6 0 + 6 = 6 So cords: (๐๐,๐๐)v โ ๐ฅ๐ฅ-intercepts (use ZERO on calc): โ๐๐,๐๐ , (๐๐,๐๐), and (๐๐,๐๐). vi โ Range: {๐๐ โ โ}.
i โ Odd degree, Positive Leading Coefficient โ So starts negative in quadrant III and ends positive in quad I.(b)ii โ Degree 5 (odd), so between 1 and 5 ๐ฅ๐ฅ-intercepts. iii โ Odd degree so no max / miniv โ Domain is {๐๐ โ โ} ๐ฆ๐ฆ-intercept is: ๐ฆ๐ฆ = (0)5+(0)4โ7 0 3 โ 13 0 2 โ 6(0) = ๐๐ So cords: (๐๐,๐๐)v โ ๐ฅ๐ฅ-intercepts: โ๐๐,๐๐ , (โ๐๐,๐๐), (๐๐,๐๐), and (๐๐,๐๐). vi โ Range: {๐๐ โ โ}.
Answers from previous page
Identifying Polynomial FunctionsClass Example 2.14
For each of the polynomial functions listed below, indicate the graph number that matches.
(a) ๐ฆ๐ฆ = ๐ฅ๐ฅ4 โ 2๐ฅ๐ฅ3 โ 7๐ฅ๐ฅ2 + 8๐ฅ๐ฅ + 12
(b) ๐ฆ๐ฆ = โ๐ฅ๐ฅ5 + 11๐ฅ๐ฅ3 + 6๐ฅ๐ฅ2 โ 28๐ฅ๐ฅ โ 24
(c) ๐ฆ๐ฆ = ๐ฅ๐ฅ5 โ ๐ฅ๐ฅ4 โ 9๐ฅ๐ฅ3 + 13๐ฅ๐ฅ2 + 8๐ฅ๐ฅ โ 12
(d) ๐ฆ๐ฆ = ๐ฅ๐ฅ4 โ 4๐ฅ๐ฅ3 โ ๐ฅ๐ฅ2 + 16๐ฅ๐ฅ โ 12
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Unit 2 โ Polynomial Functions
Applications of Polynomial Functions
In Math 20, we saw how a certain type of polynomial function, the quadratic (degree 2) function, has applications in parabolic motion and finance. (To name just two!)
Remember finding the maximum height of a ball?
Above โ classic math 20 question, do you remember how to find the max height?
i โ Starts neg. in quad. III, ends neg. IV. ii โ Can be 0 to 4 ๐ฅ๐ฅ-ints (a)2.13iii โ Even degree, neg lead coeff., so graph has MAXiv โ Domain is {๐๐ โ โ} ๐ฆ๐ฆ-intercept is: (๐๐,โ๐๐๐๐)v โ ๐ฅ๐ฅ-ints: โ๐๐,๐๐ , (๐๐,๐๐), (๐๐,๐๐). vi โ Range: {๐๐ โค ๐๐.๐๐๐๐,๐๐ โ โ}.
For RANGE, find the MAX:
(a) (b) (c) (d) 2.14
Answers from previous page
A box is with no lid is made by cutting four squares (each with a side length โ๐ฅ๐ฅโ from each corner of a 24 cm by 12 cm rectangular piece of cardboard.
(a) Determine a function that models the volume of the box.
(b) Use technology to graph the function, and sketch below. Label each axis, provide a scale, and indicate any intercepts or max / min points.Use your graphing calculator to obtain theseโฆ youโll need to โtrial-and-errorโ a suitable viewing window, indicate in your sketch below.
(c) State the domain of the function, with respect to the โreal-worldโ constraints of the problem.
(e) State the maximum volume of the box, (Round to the nearest ๐๐๐๐3)
Constructing and Analyzing a Polynomial EquationClass Example 2.15
First โ match the window to whatโs given. Graph ๐๐๐๐ = โ๐๐๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐Then โ from the CALC menu, select #4, โMAXIMUMโ
Next โ For โleft boundโ, hit anywhere to the left of the max. Do the same for โright boundโ, anywhere to the right.
The MAX should be between these arrows.
Finaly โ Hit for โGuessโ. The max value is the ๐ฆ๐ฆ-coord.
So, the maximum height of the ball is 156.24 feet, after 3.1 seconds.
(d) State the value of โ๐ฅ๐ฅโ that gives the maximum volume. (Round to the nearest hundredth)
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2.1 Characteristics of Polynomial Functions
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A box with a lid can be created by removing two congruent squares from one end of a rectangular 8.5 inch by 11 inch piece of cardboard. The congruent rectangles removed from the other end as shown. (The shaded rectangles represent the waste, or removed portions that will not be used in the box)
(a) In the diagram below there are two congruent rectangles; one that will form the base of the box, and one that will be the top. Complete the diagram by providing the missing dimensions (indicated with / ) for the base and top.
Constructing and Analyzing a Polynomial EquationClass Example 2.16
๐๐๐๐
๐๐๐๐ inches
๐๐.๐๐inches
๐๐.๐๐ inches ๐๐.๐๐ inches
(b) Determine a function that models the volume of the box.
(c) Use technology to graph the function, and sketch below. Label each axis, provide a scale, and indicate any intercepts or max / min points.Use your graphing calculator to obtain theseโฆ youโll need to โtrial-and-errorโ a suitable viewing window, indicate in your sketch below.
(d) State the domain of the function, with respect to the โreal-worldโ constraints of the problem.
(f) State the maximum volume of the box, (Round to the nearest thousandth)
(e) State the value of โ๐ฅ๐ฅโ that gives the maximum volume. (Round to the nearest thousandth)
(g) State the dimensions that yield the maximum volume. (Round to the nearest thousandth)
๐๐๐๐
๐๐ = ๐ฅ๐ฅ(12 โ 2๐ฅ๐ฅ)(24 โ 2๐ฅ๐ฅ)(a)2.15
Answers from previous page
Note that the part of the graph to the right of ๐ฅ๐ฅ = 6is โirrelevantโ, as the y-coord (Volume) is negative.
Only considerthis portion of graph
(c) Domain is [๐๐,๐๐](d) Max when ๐๐ โ ๐๐.๐๐๐๐ cm x-coord of MAX point
(e) Max Volume: โ ๐๐๐๐๐๐ ๐๐๐๐๐๐ y-coord of MAX
Remember units!
(b)(๐๐.๐๐๐๐,๐๐๐๐๐๐.๐๐๐๐)
(๐๐,๐๐)(๐๐,๐๐)
Your sketch should only be shown within the domain, [0,6]. Show dots () on the graph at the domain start & end points, just like here.
Label the coords of the intercepts and max point. Provide a scale, and label what each axis represents.
Volu
me
Length cut out
๐๐๐๐ โ ๐๐๐๐๐๐๐๐ โ ๐๐๐๐
Note: Do not expand function!
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Page|93
Unit 2 โ Polynomial Functions
2.1 Practice Questions
1. Indicate which of the following functions are polynomial functions:
(a) ๐ฆ๐ฆ = 3๐ฅ๐ฅ5 โ 3๐ฅ๐ฅ3 + 2๐ฅ๐ฅ2 + 11๐ฅ๐ฅ + 6
(d) ๐ฆ๐ฆ = 4๐ฅ๐ฅ4 โ 2๐ฅ๐ฅ2 + 5๐ฅ๐ฅโ1 โ 1
(b) ๐ฆ๐ฆ = 3๐ฅ๐ฅ3 โ 5๐ฅ๐ฅ0.5 + 2
(e) ๐ฆ๐ฆ = 3๐ฅ๐ฅ3 โ 5 ๐ฅ๐ฅ
(c) ๐ฆ๐ฆ = 5
(f) ๐ฆ๐ฆ = ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 2
2. Indicate which of the following graphs are likely those of polynomial functions:(a) (b) (c) (d)
(e) (f) (g) (h)
3. For each of the following polynomial functions, state each of the indicated characteristics. Try as many as you can without graphing.
iii - Start / end behavior
iv - Possible # of ๐ฅ๐ฅ-intercepts
v - Whether there is a max or min
vi -๐ฆ๐ฆ-intercept
i - Lead Coefficient
ii - Degree
(a) ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + 8๐ฅ๐ฅ2 + 11๐ฅ๐ฅ โ 20 (b) ๐ฆ๐ฆ = 5 โ ๐ฅ๐ฅ4
(c) ๐ฆ๐ฆ = โ2๐ฅ๐ฅ4 โ 6๐ฅ๐ฅ3 + 14๐ฅ๐ฅ2 + 30๐ฅ๐ฅ โ 36 (d) ๐ฆ๐ฆ = โ2 ๐ฅ๐ฅ + 3 2(๐ฅ๐ฅ โ 2)2(๐ฅ๐ฅ โ 1)
i -
ii -
iii -
iv -
v -
vi -
(a) i -
ii -
iii -
iv -
v -
vi -
(b) i -
ii -
iii -
iv -
v -
vi -
(c) i -
ii -
iii -
iv -
v -
vi -
(d)
๐ฝ๐ฝ = ๐๐(๐๐.๐๐ โ ๐๐๐๐)(๐๐.๐๐ โ ๐๐)
(a)2.16
Answers from previous page
(d) Domain is [๐๐,๐๐.๐๐๐๐](e) Max when ๐๐ โ ๐๐.๐๐๐๐๐๐ in x-coord of MAX point
(f) Max Volume: โ ๐๐๐๐.๐๐๐๐๐๐ ๐๐๐๐๐๐ y-coord of MAX
Remember units!
(๐๐.๐๐๐๐๐๐,๐๐๐๐.๐๐๐๐๐๐)
(๐๐.๐๐๐๐,๐๐)
Label your graph! And only show the relevant portion / within domain(c)
(g) ๐๐.๐๐๐๐๐๐ ร ๐๐.๐๐๐๐ ร ๐๐.๐๐๐๐๐๐ inches (b)
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2.1 Characteristics of Polynomial Functions
Page|94
4. For each of the following graphs, determine the indicated characteristics of the related function.
i -
ii -
iii -
iv -
v -
(a)
iii - # of ๐ฅ๐ฅ-intercepts
iv - Range
v - Constant term in function equation
i - Is the degree even or odd?
ii - Is the leading coefficient pos (+) or neg (-)
i -
ii -
iii -
iv -
v -
(b) i -
ii -
iii -
iv -
v -
(c) i -
ii -
iii -
iv -
v -
(d)
(a)(b) (c)(a)
5. For each of the following functions, use technology to determine each of the indicated characteristics. Note that using technology (graphing on your calc) is not required for each characteristic each time! For example, see if you can spot the ๐ฅ๐ฅ-intercepts of (c) without graphing. (And degree and ๐ฆ๐ฆ-ints can always be found without graphing)Also note: To get best results graphing on your calculator โ you must practice setting your window! For most of these you can use an ๐๐-min of โ๐๐ and an ๐๐-max of ๐๐. However for the ๐๐min and max โฆ. use trial and error! (Youโll want to see any relative max / min points, so ensure your window is โlarge enoughโ)
iii - The coordinates of ๐ฆ๐ฆ-intercept
iv - The Range
i - The degree
ii - The coordinates of any ๐ฅ๐ฅ-intercepts
(a) ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ3 + 8๐ฅ๐ฅ2 + 11๐ฅ๐ฅ โ 20 (b) ๐ฆ๐ฆ = ๐ฅ๐ฅ4 โ 3๐ฅ๐ฅ3 โ 12๐ฅ๐ฅ2 + 52๐ฅ๐ฅ โ 48
(c) ๐ฆ๐ฆ = โ ๐ฅ๐ฅ + 3 2(๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ โ 3) (d) ๐ฆ๐ฆ = โ2๐ฅ๐ฅ2 + 2๐ฅ๐ฅ + 24
i -
ii -
iii -
iv -
(a) i -
ii -
iii -
iv -
(b) i -
ii -
iii -
iv -
(c) i -
ii -
iii -
iv -
(d)
Note: Where applicable, round to the nearest hundredth.
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Answers to Practice Questions on the previous page
Polynomial functions are: (a), (c), (e)1. Polynomial functions are: (a), (d), and (f)2.
3. (a) i ๐๐ ii 3 iii Starts neg in quad III, ends pos in quad I iv 1 to 3 v No max or min vi (๐๐,โ๐๐๐๐)(b) i โ๐๐ ii 4 iii Starts neg in quad III, ends neg in quad IV iv 0 to 4* v Graph has a max vi (๐๐,๐๐)(c) i โ๐๐ ii 4 iii Starts neg in quad III, ends neg in quad IV iv 0 to 4 v Graph has a max vi (๐๐,โ๐๐๐๐)(d) i โ๐๐ ii 5* iii Starts pos in quad II, ends neg in quad IV iv 3* v No max or min vi (๐๐,๐๐๐๐)
see note 1
see note 2see note 3
Note 1: We can visualize this, as the graph of ๐ฆ๐ฆ = ๐ฅ๐ฅ4 is similar to ๐ฆ๐ฆ = ๐ฅ๐ฅ2, so visualize a โparabolaโ opening down and shifted5 units up. So we know, without graphing, that there will be TWO ๐ฅ๐ฅ-intercepts!
Note 2: For functions in factored form, the degree of the entire function is the sum of all exponents, so: 2 + 2 + 1 = ๐๐๐ฆ๐ฆ = โ2 ๐ฅ๐ฅ + 3 ๐๐(๐ฅ๐ฅ โ 2)๐๐(๐ฅ๐ฅ โ 1)
Note 3: Each factor corresponds to one ๐ฅ๐ฅ-intercept, so we know with certainty there are 3. Thereโs an invisible โ1โ here!
Page|95
Unit 2 โ Polynomial Functions
Answers to Practice Questions on the previous page
4. (a) i ODD ii NEGATIVE iii 4 ๐ฅ๐ฅ-intercepts iv {๐๐ โ โ} v Constant term: โ๐๐๐๐ (represented by ๐ฆ๐ฆ-intercept)(b) i EVEN ii POSITIVE iii 3 ๐ฅ๐ฅ-intercepts iv {๐๐ โฅ โ๐๐๐๐.๐๐๐๐,๐๐ โ โ} v Constant term: โ๐๐(c) i ODD ii POSITIVE iii 2 ๐ฅ๐ฅ-intercepts iv {๐๐ โ โ} v Constant term: โ๐๐๐๐(d) i EVEN ii NEGATIVE iii 3 ๐ฅ๐ฅ-intercepts iv {๐๐ โค ๐๐๐๐.๐๐๐๐,๐๐ โ โ} v Constant term: ๐๐
(a) i ๐๐ ii โ๐๐,๐๐ , (โ๐๐,๐๐) and (๐๐,๐๐) iii (๐๐,โ๐๐๐๐) iv {๐๐ โ โ}(b) i ๐๐ ii โ๐๐,๐๐ , (๐๐,๐๐) and (๐๐,๐๐) iii (๐๐,โ๐๐๐๐) iv {๐๐ โฅ โ๐๐๐๐๐๐.๐๐๐๐,๐๐ โ โ}(c) i ๐๐ ii โ๐๐,๐๐ , (๐๐,๐๐) and (๐๐,๐๐) iii (๐๐,โ๐๐๐๐) iv {๐๐ โค ๐๐๐๐.๐๐๐๐,๐๐ โ โ}
Note: Each factor provides an ๐ฅ๐ฅ-intercept(d) i ๐๐ ii โ๐๐,๐๐ and (๐๐,๐๐) iii (๐๐,๐๐๐๐) iv {๐๐ โค ๐๐๐๐.๐๐,๐๐ โ โ}
5.
6. Without graphing (use your reasoning abilities!), match each of the following functions with its graph.
(a) ๐ฆ๐ฆ = โ๐ฅ๐ฅ5 + 12๐ฅ๐ฅ3 + 2๐ฅ๐ฅ2 โ 27๐ฅ๐ฅ โ 18
๐ฆ๐ฆ = โ๐ฅ๐ฅ4 + ๐ฅ๐ฅ3 + 11๐ฅ๐ฅ2 โ 9๐ฅ๐ฅ โ 18(b)
๐ฆ๐ฆ = ๐ฅ๐ฅ5 โ 2๐ฅ๐ฅ4 โ 10๐ฅ๐ฅ3 + 20๐ฅ๐ฅ2 + 9๐ฅ๐ฅ โ 18(c)
๐ฆ๐ฆ = โ๐ฅ๐ฅ4 + ๐ฅ๐ฅ3 + 7๐ฅ๐ฅ2 โ 13๐ฅ๐ฅ + 6(d)
7. A package may be sent through a particular mail service only if it conforms to specific dimensions. To qualify, the sum of its height plus the perimeter of its base must be no more than 72 inches. Also for our design, the base of the box (shaded in the diagram below) has a length equal to double the width.
base
๐๐ =
2๐ฅ๐ฅ๐ฅ๐ฅ
(a) In the blank on the left, state an expression for the height (๐๐) of the box.Need a hint? See the bottom of the next page.
(b) Determine a function that represents the Volume of the box.
(c) Use technology to graph the function obtained in (b) with a suitable viewing window. Provide your sketch below, labeling any max/mins and intercepts. Also fully label the axis, what each axis represents, and a suitable scale.
(d) Provide a domain and range for your function obtained in (b), with respect to the โreal worldโ constraints of the problem.
Domain: Range:
(e) State the maximum volume of the box that can be sent.
(f) State the dimensions for the box that provides the maximum volume.
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2.1 Characteristics of Polynomial Functions
Page|96
๐ฝ๐ฝ = (๐๐๐๐)(๐๐)(๐๐๐๐ โ ๐๐๐๐)
(a)7.
Answers from previous and this page
(a) (b) (c) (d)
The perimeter of the base is: 2๐ฅ๐ฅ + 2๐ฅ๐ฅ + ๐ฅ๐ฅ + ๐ฅ๐ฅ = ๐๐๐๐. As we wish for the largest volume box, weโll use all 72 inches (sum of perimeter and height) available. So โ + 6๐ฅ๐ฅ = 72, and ๐๐ = ๐๐๐๐ โ ๐๐๐๐.
HINT for #7(a):
(d) Domain is [๐๐,๐๐.๐๐๐๐]
Max when ๐๐ = ๐๐ inches (f) Max Volume: = ๐๐๐๐๐๐๐๐ ๐๐๐๐๐๐
(g) ๐๐๐๐ length ร ๐๐ width ร ๐๐๐๐ height inches
(๐๐,๐๐๐๐๐๐๐๐)
(๐๐๐๐,๐๐)(๐๐,๐๐)
Volu
me
width of box
Range is [๐๐,๐๐๐๐๐๐๐๐]
Graph ๐๐๐๐ = in your calculator. Trial-and-error to get best window. Sketch should only show graph within domain. (between 0 and 6)
6.
๐๐ = ๐๐๐๐ โ ๐๐๐๐(c)
(b)
8. An open box is to be made by cutting out squares from the corners of an 8 inch by 15 inch rectangular sheet of cardboard and folding up the sides.
(a) On diagram 1 on the right, provide expressions that represent the length and width of the finished box.
(c) Use technology to graph the function, and sketch below. Label each axis, provide a scale, and indicate any intercepts or max / min points. Use your graphing calculator, provide a sketch below.
(d) State the domain and range of the function, with respect to the โreal-worldโ constraints.
(f) State the maximum volume of the box, (Round to the nearest ๐๐๐๐3)
(e) State the value of โ๐ฅ๐ฅโ that gives the maximum volume. (Round to the nearest hundredth)
(b) Determine a function that models the volume of the box.
Diagram 1 Diagram 2
(g) Provide the dimensions that yield the box of maximum volume, (Round to the nearest hundredth)
๐ฝ๐ฝ = (๐๐๐๐ โ ๐๐๐๐)(๐๐ โ ๐๐๐๐)(๐๐)
(a)8.(d) Domain is [๐๐,๐๐]
(e) Max when ๐๐ = ๐๐.๐๐๐๐ inches
(f) Max Volume: = ๐๐๐๐.๐๐๐๐ ๐๐๐๐๐๐
(g) ๐๐๐๐.๐๐๐๐ length ร ๐๐.๐๐๐๐ width ร ๐๐.๐๐๐๐height inches
Range is [๐๐,๐๐๐๐.๐๐๐๐]
(๐๐.๐๐๐๐,๐๐๐๐.๐๐๐๐)
(๐๐,๐๐)(๐๐,๐๐)
Volu
me
Height
(b)
(c)
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