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This is the second of our many quizzes on graphing polynomials with a lot of graphs of polynomial questions. Click on this to access more such competitive quizzes on polynomials.

Graphing Polynomials Quiz 2

mathematics

 Which of the following graphs represents an even degree polynomial?

 Check the end graphed lines of each graph. 

  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/441f9eef-8a71-4e8a-a81f-e0dffcc5cda27675793665747797334.png"/>$ \\ $ In an even degree polynomial, ends of the graphed lines face the same direction either up or down while in odd degree polynomial ends of the graphed lines face the opposite direction.

 What can we say about the degree and the leading coefficients of the following graph?<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/829a0930-99fd-458b-acde-6f0ee3a245a01924379810051001443.png"/>

 For leading coefficient check whether the right side of a graph is facing up or down and calculate the number of turning points for a degree of a polynomial.

  A polynomial graph has at most $(n – 1)$ turns, where n is the possible degree of the polynomial. Here, we have 4 turns in the graph so the least possible degree of a polynomial is 5. Also, right side of the graph facing down so leading coefficient i.e., $a < 0$. 

 Find the vertex of polynomial $y = f(x)$ whose graph is given below.<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/855af378-13f7-4f14-a44f-b6265c243e367596858819426109510.png"/>

 First find the zeros to form a quadratic polynomial. Now use the vertex formula to find the required answer.

 $\text{From the above graph, we get that -5 and 15 are the two zeros.} \\ \text{Required polynomial;} \\ \Rightarrow (x+5)(x-15) \Rightarrow {{x}^{2}}-10x-75 \\ \text{Required Vertex;} \\ \Rightarrow \left( \dfrac{-b}{2a},\dfrac{-D}{4a} \right) \\ \Rightarrow \left( \dfrac{10}{2(1)},\dfrac{-400}{4(1)} \right)\text{       }(\because  D={{b}^{2}}-4ac={{(-10)}^{2}}-4(1)(-75)) \\ \Rightarrow (5,-100)$ 

 Which of the following polynomials can be formed from the given graph?<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/fd6c92ed-6f04-4826-aa27-aa626f41175b3903967190345056567.png"/>

 Find the zeros with their multiplicity and then multiply all the factors to get a required polynomial.

 $\text{Here, we have the zeros and multiplicities as follows:} \\ x = -5, \text{ Multiplicity 1 because it crosses the x-axis} \\ x = 0, \text{ Multiplicity 2 because it just touches the x-axis but not crosses the x-axis} \\ x = 2, \text{ Multiplicity 1 because it crosses the x-axis} \\ \Rightarrow \text{ Required polynomial, } (x+5){{(x+0)}^{2}}(x-2) \\ \Rightarrow ({{x}^{2}}+3x-10)({{x}^{2}}) \\ \Rightarrow {{x}^{4}}+3{{x}^{3}}-10{{x}^{2}}$ 

 Find the coefficients $l, m, n, o$ and $p$ of the polynomial $y=l{{x}^{4}}+m{{x}^{3}}+n{{x}^{2}}+ox+p$ whose graph is given below?<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/66b3b42f-2576-44bb-8aad-9ee582ff87187415050155385510045.png"/>

 Find the zeros with their multiplicity and then form a polynomial. Now, find the value of $k$ using y-intercept and then compare the coefficients.

 $\text{Here, the zeros are, }x = -1\text{ and }x = 3\text{ with multiplicity 1 as the graph crosses the x-axis.} \\ x = 1\text{ with multiplicity 2 as it just touches the x-axis but not crosses the x-axis.} \\ \text{Required polynomial }=y=k(x+1)(x-3){{(x-1)}^{2}} \\ \Rightarrow k({{x}^{4}}-4{{x}^{3}}+2{{x}^{2}}+4x-3) \\ \text{y – intercept of a polynomial is }(0, -3). \\ \Rightarrow -3=k({{(0)}^{4}}-4{{(0)}^{3}}+2{{(0)}^{2}}+4(0)-3) \\ \Rightarrow k=1 \\ \text{On comparing with the coefficients of }l{{x}^{4}}+m{{x}^{3}}+n{{x}^{2}}+ox+p,\text{ we get, }\\ l = 1, m = -4, n = 2, o = 4, p = -3$