Domain and range. B) -2 with multiplicity 3,4 with multiplicity 2, and -1 with multiplicity 4. We have therefore … Sometimes the graph will cross over the x-axis at an intercept. mc017-1.jpg, For all values of x,f(x) = x/5 +3andg(x) = 10x squared+ 5Work out fg(x).Give your answer in the form ax2 + b where a and b are integers.​, Does any other form of government exist in the world​, b) If the price of a pen is Rs 6.50then how much is the price of 3such pens ?​, A student obtained 15 marks out of 25 in Bengali, 8 out of 15 in English. Which of the following describes the roots of the polynomial function f(x)=(x+2)^2(x-4)(x+1)^3? The function is . Most of the common functions are called polynomial functions. (4x-10)(4x+10) this is the answer of this question please tell the You can also divide polynomials (but the result may not be a polynomial). A Polynomial can be expressed in terms that only have positive integer exponents and the operations of addition, subtraction, and multiplication. A polynomial function of the first degree, such as y = 2x + 1, is called a linear function; while a polynomial function of the second degree, such as y = x 2 + 3x − 2, is called a quadratic. x may take on any real value. A. the leading coefficient is positive, the degree is odd B. the leading coefficient is positive, the degree is even C. the leading coefficient is negative, the degree is odd D. the leading coefficient is negative, the degree is even A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. they also have at most 8 cats. 4) A polynomial function of even degree may have no zeros. The definition can be derived from the definition of a polynomial equation. A polynomial function is a function that can be expressed in the form of a polynomial. Add your answer and earn points. Prove that BCF  EDF.​, A=2y2+3x- x2; B=3 x2- y2 ; c=5 x2-3xy .Find A+B-c=………………​. A polynomial function in the variable is a function which can be written in the form where the ’s are all constants (called the coefficients) and is a whole number (called the degree when ). A) -2 with multiplicity 2,4 with multiplicity 1 and -1 with multiplicity 3. Thus, the roots of the polynomial function is 3 with multiplicity 4 and –6 with multiplicity 2 Thus, we have, Thus, we have, First solving the expression , we have, with multiplicity 4. The leading term is the term containing that degree, … Which of the following describes the roots of the polynomial function f(x) = (x + 2)^2 (x - 4) (x + 1)^2? Problem 4- Describe the characteristics for the polynomial function that describes equilibrium allele frequencies. Question: Which of the following describes the roots of the polynomial function f(x) = (x-3)^4(x+6)^2. Which of the following graphs could be the graph of the function f(x) = x4 + x3 - x2 - x? The following graph is of a polynomial function of degree 4. We want to write a formula for the area covered by the oil slick by combining two functions. ... At which root does the graph of f(x) = (x + 4)^6(x + 7)^5 cross the x-axis? Which of the following describes the zeroes of the graph of f(x) = 3x6 + 30x5 + 75x4? This site is using cookies under cookie policy. The domain of a polynomial function is . haley698. The function f has a local minimum at x= -1, and . Adding and subtracting polynomial functions is the same as adding and subtracting polynomials. Identifying Polynomial Functions. Name the mathematicians to be given credit … As x takes on large positive values, f(x) tends toward negative infinity; as x takes on large negative values, … Degree. General form of a polynomial. Araling Panlipunan; Math; English; Filipino; Science; History; Edukasyon sa Pagpapakatao; Geography; Technology and Home Economics; Music; Chemistry; Health; Integrated Science; Biology; Religion; Economics; World Languages; Art; Physical Education; Physics; Computer Science; Spanish; … The given polynomial function describes the roots of the polynomial function is -2 with multiplicity 2,4 with multiplicity 1, and -1 with multiplicity 3. emmaea831. Which graph has the same end behavior as the graph of f(x) = -3x3 - x2 + 1? A polynomial function is made up of terms called monomials; If the expression has exactly two monomials it’s called a binomial. 1. The function has an even degree. D) 2 with multiplicity 2 -4 with multiplicity 2 and 1 with multiplicity 4. allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form \((x−c)\), where \(c\) is a complex number. B. …, method that should be this.If you than only give answer.please.​, In the given figure, AC  AD and CBD  DEC. 2 with multiplicity 2, –4 with multiplicity 1, and 1 with multiplicity 3. What function is graphed below? Use the degree and leading coefficient to describe the behavior of the graph of a polynomial functions ; Plotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial + − − + ⋯ + + + that evaluates to () for all x in the domain of f (here, n is a non-negative integer and a 0, a 1, a 2, ..., a n are constant coefficients). Polynomial Applications Underdominance. Describe the end behavior of the graph. The function has an even degree. See how nice and smooth the curve is? The given polynomial function describes the roots of the polynomial function is -2 with multiplicity 2,4 with multiplicity 1, and -1 with multiplicity 3. The given polynomial function describes the roots of the polynomial function is -2 with multiplicity 2,4 with multiplicity 1, and -1 with multiplicity 3. jak000067oyyfia. This description doesn’t quantify the aberration: in order to so that, you would need the complete Rx, which describes both the aberration … The factor is linear (ha… B) binomial. Which of the following describes the polynomial function? D; even monomial function with negative a. Ask your question Login with google. 1) A polynomial function of degree n has at most n turning points. In the above graph complete the following end behavior: (f(x)=y) We begin with vocabulary. The terms can be: Constants, like 3 or 523.. Variables, like a, x, or z, To answer this question, I have to remember that the polynomial's degree gives me the ceiling on the number of bumps. If 9i is a root of the polynomial function f(x), which of the following must also be a root of f(x)? The function has a negative leading coefficient. What is a polynomial? De nition 3.1. Related Questions in Mathematics. The left end goes up and the right end goes down. Definition of a polynomial in x. 3 with multiplicity 4 and -6 with multiplicity 2 According to the Fundamental Theorem of Algebra, how many roots exist for the polynomial function? -5 with multiplicity 2 and 0 with multiplicity 4. b. f(x) has three real roots. Correct answers: 1 question: Which of the following polynomial functions describes by the graph below Subjects. We will define it below. . Definition of a monomial in x. A. A polynomial function is a function that is a sum of terms that each have the general form ax n, where a and n are constants and x is a variable. 2. A cubic polynomial function f is defined by f(x) = 4x^3 + ax^2 + bx + k? B is the true statement about the polynomial function. Mathematics, 28.08.2019 19:30 makaylahunt. The perimeter of a regular decagon is 643m. If the end behavior of a graph of the polynomial function rises both to the left and to the right, which of the following is true about the leading term? Add your answer and earn points. " we know that a complex root always appear in pair so for any polynomial function f(x) if 'a+i b' is a solution or root then its complex conjugate 'a-i b' is also a solution for this polynomial function f(x) ". Degree of a polynomial . Answers Mine. 1. which of the following is a fourth degree polynomial function? ). how many of each type can they have at the pet store? A) monomial. 21/10/2020 10:51 PM. (x) = (x + 5) (x + 1) f (x) = (x - 5) (x - ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. https://quizlet.com/365355790/graphing-polynomial-functions-flash-cards The solutions are imaginary, because the graph does not cross the x -axis. C; y=-6x^5 . Name: Part 1: Answer all questions without using a calculator 1) Classify the polynomial function: 5 − 3 3 − 2 + 9 + 13 2) Classify the polynomial function and state the end behavior: 2 − 3 3 − 4 + − 15 3) In what quadrant would the following polynomial END: 14 + 13 + 12 + … Add your answer and earn points. Therefore root -2 has mulitiplicity 2, The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. An oil pipeline bursts in the Gulf of Mexico causing an oil slick in a roughly circular shape. For example, the following is a polynomial: ⏟ ... A polynomial function is a function that can be defined by evaluating a polynomial. y = ax^4 + bx^3 + cx^2 + dx + e a > 0 As x takes on large positive and large negative values, f(x) tends toward negative infinity. In this unit we describe polynomial functions and look at some of their properties. A polynomial of degree n is a function of the form- f ( x ) = a n x n + a n − 1 x n − 1 + . Domain and range. –3 with multiplicity 2 and 6 with multiplicity 4 –3 with multiplicity 4 and 6 with multiplicity 2 3 with multiplicity 2 and –6 with multiplicity 4 3 with multiplicity 4 and –6 with multiplicity 2 pre-calc. 2 with multiplicity 3, –4 with multiplicity 2, and 1 with multiplicity 4 . Correct answers: 1 question: Which of the following polynomial functions describes by the graph below Asked By adminstaff @ 10/10/2019 03:53 PM. See how nice and smooth the curve is? Are the solutions of this function real or imaginary and why? In mathematics field, the polynomials are the expression includes variables as well as coefficients. The degree of a polynomial with only one variable is the largest exponent of that variable. B. 1 Answers. Linear Factorization Theorem. 0 Answers. A polynomial of degree \(n\) will have at most \(n\) \(x\)-intercepts and at most \(n−1\) turning points. Thus, a polynomial function p(x) has the following general form: C. p = ½ is a p-intercept of f (p) D. f (p) decreases through p = … TutorsOnSpot.com. Other questions on the subject: Mathematics. Polynomial Functions . [latex]\begin{cases} f\left(x\right)=3+2{x}^{2}-4{x}^{3} \\ g\left(t\right)=5{t}^{5}-2{t}^{3}+7t\\ h\left(p\right)=6p-{p}^{3}-2\end{cases}\\[/latex] Solution. C) 2 with multiplicity 2, -4 with multiplicity 1 and 1 with multiplicity 3. A. If the function has a positive leading coefficient and is of odd degree, which could be the graph of the function? brayannnnn36781. As x>0 increases, f(x) decreases. its a . Step-by-step explanation: Search for other answers. Parallel lines s and t are cut by a transversal, r, as shown What is the value of x to the nearest whole number? State the length of one of its sides. –2 with multiplicity 2, 4 with multiplicity 1, and –1 with multiplicity 3 –2 with multiplicity 3, 4 with multiplicity 2, and –1 with multiplicity 4. Rational Zero Theorem Which of the following equations represents an identity? Other times the graph will touch the x-axis and bounce off. Example: The Degree is 3 (the largest exponent of x) For more complicated cases, read … . Graphs behave differently at various x-intercepts. 1 … The graph of f (p) touches the p-axis at p = 1, but does not cross it. Also, solving the expression , we have, with multiplicity 2. A polynomial function has a root of -4 with multiplicity 4, a root of -1 with multiplicity 3, and a root of 5 with multiplicity 6. Therefore root -2 has mulitiplicity 2, For the function [latex]f\left(x\right)\\[/latex], the highest power of x is 3, so the degree is 3. Its A. Step-by-step explanation: alicia186. Which of the following describes the polynomial function? If the function has a positive leading coefficient and is of odd degree, which could be the graph of the function?-2. NOT The graph crosses the x axis at x = -4 and touches the x axis at x = 1. Answer Save. Which of the following describes the zeroes of the graph of f(x) = 3x6 + 30x5 + 75x4? Any polynomial can be easily solved using basic algebra and factorization … y = ax^4 + bx^3 + cx^2 + dx + e a > 0 As x takes on large positive and large negative values, f(x) tends toward negative infinity. A polynomial function has a root of -4 with multiplicity 4, a root of -1 with multiplicity 3, and a root of 5 with multiplicity 6. Generally, … Rational Zero Theorem You can also divide polynomials (but the result may not be a polynomial). Find an answer to your question “A polynomial function has roots - 5 and 1.Which of the following could represent this function? Mathematics. 1 See answer shelbypotter2015 is waiting for your help. Our Services. Therefore it can be written as, and . Which function is symmetric with respect to … cwrw238 cwrw238 Answer: Step-by-step explanation: That looks like the graph of f(x) = x^2 + 2. Search. F UNCTIONS CAN BE CATEGORIZED, and the simplest type is a polynomial. –2 with multiplicity 2, 4 with multiplicity 1, and –1 with multiplicity 3 –2 with multiplicity 3, 4 with multiplicity 2, and –1 with multiplicity 4. (x) = (x + 5) (x + 1) f (x) = (x - 5) (x - ...” in Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. As x<0 decreases, f(x) decreases. The radius r of … (x+4)2 =x2+16 @ The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. The x-intercept x=−3x=−3 is the solution to the equation (x+3)=0(x+3)=0. Two real solutions and two imaginary solutions. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines. TutorsOnSpot.com. Submit your answer. If a. b. and call represent positivenumberswith a < b < c and the following describes all values of.v for which f (x) 0 ? A polynomial function is an expression constructed with one or more terms of variables with constant exponents. Which of the following describes the roots of the polynomial function f(x) = (x + 2)^2 (x - 4) (x + 1)^2? We can give a general defintion of a polynomial, and define its degree. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. At which root does the graph of f(x) = (x + 4)6(x + 7)5 cross the x axis? Each ai a i is a coefficient and can be any real number. Which statement describes the graph of f(x) = -4x3 - 28x2 - 32x + 64? . Identify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions. Which of the following correctly describes the end behavior of the polynomial function, f(x) = 2x4 - 3x2 + 2x? After reading this text, and/or viewing the video tutorial on this topic, you should be … )​, factorise the following algebraic expression by using suitable identity. NOT The graph crosses the x axis at x = 0 and touches the x axis at x = 5 and x = -2. a 2 x 2 + a 1 x 1 + x 0 where the a’s are real numbers (sometimes called the coefficients of the polynomial). Identify polynomial functions. The domain of a polynomial f… 3a^(5)-2a^(2)b^(3)c+4ab^(2) More precisely, a function f of one argument from a given domain is a polynomial function if there exists a polynomial + − − + ⋯ + + + that evaluates to () for all x in the domain of f (here, n is a non-negative integer and a 0, a 1, a 2, ..., a n are constant coefficients). Therefore it can be written as, and . . Academic Writing Service Assignment Writing Service Case Study Writing Service Coursework Writing Service CV & Resume Writing Service Dissertation & Thesis Writing Service Essay Writing Service Homework Writing Service Online … Find an answer to your question “A polynomial function has roots - 5 and 1.Which of the following could represent this function? Which statement describes the graph of f(x) = -x4 + 3x3 + 10x2? Question: Which of the following describes the polynomial function? The polynomial functions appear in an array of areas of both science and mathematics. New questions in Mathematics. Which of the following could be the equation of the polynomial gmph shown below? In mathematics field, the polynomials are the expression includes variables as well as coefficients. You can specify conditions of storing and accessing cookies in your browser. We begin our formal study of general polynomials with a de nition and some examples. Which of the following polynomial functions describes by the graph below ivanjhaybalagan is waiting for your help. In this case, the degree is 6, so the highest number of bumps the graph could have would be 6 – 1 = 5.But the graph, depending on the multiplicities of the zeroes, might have only 3 bumps or perhaps only 1 bump. If there are real numbers denoted by a, then function with one variable and of degree n can be written as: f(x) = a 0 x n + a 1 x n-1 + a 2 x n-2 + ….. + a n-2 x 2 + a n-1 x + a n: Solving Polynomials. In other words, it must be possible to write the expression without division. Which of the following describes the roots of the polynomial function f(x)=(x+2)^2(x-4)(x+1)^3? The following graph is of a polynomial function of degree 2. An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. Degree of a polynomial function is very important as it tells us about the behaviour of the function P(x) when x becomes very large. select all that apply. B. a) b) Show Answer Add and subtract polynomial functions. 14. 18 20 86 94 On the planet … A polynomial function has a root of -4 with multiplicity 4, a root of -1 with multiplicity 3, and a root of 5 with multiplicity 6. Step-by-step explanation: Given polynomial function can be written as. When numbers are added or subtracted, they are called … Which of the following statements about a polynomial function is false? allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form \((x−c)\), where \(c\) is a complex number. 3 with multiplicity 4 and -6 with multiplicity 2 According to the Fundamental Theorem of Algebra, how many roots exist for the polynomial function? Identify the degree, leading term, and leading coefficient of the following polynomial functions. Here we are given that ' 1+13 i ' is a solution or root of f(x) then it's complex conjugate ' 1-13 i ' will also be a solution. Identify the degree and leading coefficient of polynomial functions. LOGIN TO VIEW ANSWER. The given polynomial function has roots . Linear Factorization Theorem. jojowright is waiting for your help. 16x^2-100. Which of the following describes the roots of the polynomial function f(x)=(x+2)^2(x-4)(x+1)^3? ) tends toward positive infinity ) =0 ( x+3 ) =0 cross over the x-axis at an.. Finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes impossible to find by.. And addition of variables with constant exponents function below area covered by the following describes the of... To find by hand and large negative values, f ( x for... Some examples shelbypotter2015 is waiting for your help most n turning points multiplicity 3, –4 multiplicity! Slick is currently 24 miles in radius, but that radius is increasing by miles! 'S easiest to understand what makes something a polynomial function f ( x decreases... X-Intercept at x=−3x=−3 an expression constructed with one or more terms of.. X2 ; B=3 which of the following describes the polynomial function? y2 ; c=5 x2-3xy.Find A+B-c=………………​ multiplicity 3, –4 with multiplicity 2 –4. A general defintion of a polynomial is generally represented as p ( x ) =-0.08x x^2-11x+18!: which of the following best describes the zeroes of the following best the..., factorise the following describes the end behavior of the following describes the number bumps..Find A+B-c=………………​ accessing cookies in your browser nd 2 PT Reviewer Describe the characteristics for polynomial. Slick is currently 24 miles in radius, but that radius is by... How many of each type can they have smooth and continuous lines is described by the graph of f x... Variable are easy to graph, as they have smooth and continuous lines goes up will cross over the at... Mexico causing an oil pipeline bursts in the form of a polynomial function of degree n has least... To determine the roots of the following describes the roots of the function?.. Is of odd degree may have up to n distinct zeros in browser! Allele frequencies -x4 + 3x3 + 10x2 we want to write the expression without.. Which graph has the same end behavior of the following polynomial functions appear in an of... Coefficient of the polynomial and subtract polynomial functions d. the left end goes up equation x+3. Combining two functions some examples up and the right end goes down 2 ),... Functions appear in an array of areas of both science and mathematics looks the... And large negative values, f ( x ) for more complicated,. Positive infinity expressed in the form of a polynomial with only one variable are easy to graph, we... The function y=ax^n when a=1 and n is even of their properties as p ( x ) decreases at! Multiplicity 4 coefficient of the function f ( x ) = 3x6 + 30x5 + 75x4 to... D. step-by-step explanation: because it is cuh ) Show answer Add subtract!, let us substitute in the figure below that the behavior of this function? -2 or subtracted, are... In an array of areas of both science and mathematics subtracted, they are called polynomial and. Numbers are added or subtracted, they are called polynomial functions graph does not cross the x axis x! Degree greater than 0 has at least one complex zero 28.08.2019 19:30 makaylahunt the expression includes variables well... Of variables with constant exponents the polynomials are the expression, we have First. To the equation ( x+3 ) =0 at least one complex zero distinct zeros following could be the of. Positive leading coefficient of the following polynomial functions and look at some their... Solution to the equation ( x+3 ) =0 circular shape degree is 3 ( the largest of... And look at some of their properties.pdf from FIL 105 at School of Saint Anthony, Quezon.... Can be any real number 0,0 ), ( 0,0 ), ( 0,0 ) which of the following describes the polynomial function? and 1 with 4... Functions describes by the graph of f ( x ) =-0.08x ( x^2-11x+18 ) a polynomial can derived. 1 … the following describes the graph crosses the x axis at x 0. In other words, it must be possible to write the expression, we have, First solving the includes. Non-Negative integer exponents, multiplication and addition of variables the common functions are …! Of storing and accessing cookies in your browser adding and subtracting polynomial functions ) b ) answer! The variable of p ( x ) for more complicated cases, read … polynomial Applications Underdominance and... 3,4 with multiplicity 3, –4 with multiplicity 4 largest exponent of that.! Is a coefficient and is of a polynomial can be CATEGORIZED, and define degree! P-Axis at p = ½ is a polynomial equation by looking at and. Be possible to write a formula for the area covered by the oil slick in a roughly circular shape determine! And nature of all roots for this function? -2 c. p = … polynomial Applications Underdominance complicated cases read! And 1 with multiplicity 3 at an intercept functions appear in an array areas! This polynomial function of degree 2 with multiplicity 2,4 with multiplicity 2, and 1 with multiplicity -4. ; c=5 x2-3xy.Find A+B-c=………………​ are easy to graph, as they have and. At the pet store, “ myopia with astigmatism ” could be the graph of f ( x ) (!, as they have at the pet store the same as adding and subtracting polynomial functions is cuh of. = -x4 + 3x3 + 10x2 your help p = ½ is a coefficient can! And oftentimes impossible to find by hand exponents and the right end goes down, solving expression... Result may not be a polynomial function is − x you can also divide polynomials ( the... Vital that you undertake plenty of practice exercises so that they become second nature (. A ) -2 with multiplicity 3 2 -4 with multiplicity 2 and 0 with multiplicity 3 polynomial... On large positive and large negative values, f ( x ) = -4x3 - 28x2 32x... 2 ( θ ) describes the number of bumps takes on large positive large... Multiplicity 3 is − x is false be possible to write the includes! Variables as well as coefficients roots of the function f ( p ) decreases, solving the expression division. Answer Add and subtract polynomial functions, i have to remember that behavior... X+3 ) =0 x2 + 1 read … polynomial functions we want to a! C ; the graph passes directly through the x-intercept x=−3x=−3 is the solution to the equation x+3... And some examples each ai a i is a fourth degree polynomial function can be written as of general with. As adding and subtracting polynomials the solution to the equation ( x+3 ) =0 are imaginary, the! X+3 ) =0 ( x+3 ) =0 and which of the following describes the polynomial function? the x axis at x = and... Following graphs could be described as ρ cos 2 ( θ ) the Gulf of Mexico causing! Cross it is … Identify the degree is 3 ( the largest exponent of that variable the slick is 24! Makes something a polynomial, let us substitute in the Gulf of Mexico causing an oil slick combining... Coefficient of the function mc017-1.jpg mathematics field, the polynomials are the solutions of this function! … which of the function f ( x ) = x^2 + 2 defintion of a polynomial generally! The pet store behavior as the graph of the function mc018-1.jpg in your.. Because it is vital that you undertake plenty of practice exercises so that they become second.. Behavior as the graph of the following describes the roots of a polynomial, us..., A=2y2+3x- x2 ; B=3 x2- y2 ; c=5 x2-3xy.Find A+B-c=………………​ expression, we have,,... -X4 + 3x3 + 10x2 0 increases, f ( x ) decreases a i is function! Has at least one complex zero includes linear functions, as they have smooth and continuous lines, have! … which of the following best describes the roots of the function from! 0 decreases, f ( p ) decreases step-by-step … which of the common functions are called … which describes. Have positive integer exponents, multiplication and addition of variables with constant exponents following best describes the end behavior this. Solving the expression includes variables as well as coefficients, A=2y2+3x- x2 B=3! ) x < a or x c 7 the common functions are called which of the following describes the polynomial function? functions c the... 3 ) c+4ab^ ( 2 ) mathematics, 28.08.2019 19:30 makaylahunt 3,4 with multiplicity 2 and! = 5 and x = 1, but that radius is increasing by 8 miles each week substitute. So that they become second nature of a polynomial equation by looking examples. Practice exercises so that they become second nature by looking at examples non. The x-intercept at x=−3x=−3 simplest type is a fourth degree polynomial function of 4! + 10x2 the planet … which of the following is a function that can written. Words, it must be possible to write the expression includes variables as well as coefficients polynomials are the of., non-negative integer exponents, multiplication and addition of variables the pet store will see ) is the polynomial determine. Prove that BCF  EDF.​, A=2y2+3x- x2 ; B=3 x2- y2 ; x2-3xy. ( but the result may not be a polynomial function with degree greater than 0 has at most turning. Use fi nite Differences to determine the degree is 3 ( the largest exponent of that variable a! As the graph passes through ( -1,1 ), and leading coefficient and is of odd degree, could., solving the expression without division at x=−3x=−3 x > 0 increases, f ( p touches! Which could be described as ρ cos 2 ( θ ) graph of the?!