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This is the second of our many quizzes on quadratic equation graphs. Click on this to access more such competitive quizzes on all topics of Quadratics.

Graphing Quadratics Quiz 2

mathematics

 Determine the sum of roots of the quadratic equation sketched below.<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/b22edb30-d30d-43d7-a75d-6b49ca7cd6d78411890949322029318.png"/>

 Find the roots from the given graph and add them.

 We know that the roots of the quadratic equation are the x-intercepts. So, we have the roots as -1 and 5.<br>Therefore, sum of roots $= -1 + 5 = 4$.

 Draw the graph for $3{{x}^{2}}-20x+12=0$. In which quadrant would the vertex lie?

 Use factorisation method to find the roots. Get the coordinates of the vertex using roots and the given equation.

 $\begin{align} & 3{{x}^{2}}-18x-2x+12=0 \\ & \Rightarrow \left( 3x-2 \right)\left( x-6 \right)=0 \\ & \Rightarrow x=\dfrac{2}{3},6 \\ & \text{x-coordinate of vertex of parabola }=\dfrac{\dfrac{2}{3}+6}{2} = \dfrac{20}{6} \approx 3.33 \\ & \text{y-coordinate of vertex of parabola }=3{{\left( \dfrac{20}{6} \right)}^{2}}-20\left( \dfrac{20}{6} \right)+12\approx -21.33 \\ & \text{Hence the vertex lies in the fourth quadrant }\\ & \\ & \end{align}$<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a32d7849-f227-4292-bfcf-766cc36ec98a7662327513971153957.png"/>

 Arya buys some pens at Rs. 60. If she had bought 2 more pens for the same amount, each pen would have cost Rs. 1 less. Solve graphically to find the number of pens she had bought?

 Using the given conditions and unitary method, form the quadratic equation. Find values of $y$ for $x = -2, -1, 0, 1, 2$ and plot graphs.

 $\begin{align} & \text{Let the number of pens bought be “x”, Cost of “x” pens }= Rs. 60 \\ & \text{So, we have a cost of 1 pen }= Rs. \dfrac{60}{x}, \\ & \text{New number of pens }= x + 2 \\ & \text{New cost of 1 pen }= Rs. \dfrac{60}{x+2} \\ & \text{On Applying condition, we get}\dfrac{60}{x}-\dfrac{60}{x+2}=1 \\ & \text{Taking LCM and simplifying, we get }{{x}^{2}}+2x-120=0 \\ & \text{ For }x = -2, \text{ we have }y=4-4-120=-120 \\ & \text{Taking values of “x” as 0, 1, 2, we have} \\ & \\ & \end{align}$<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/3f6a48fe-d098-4ae7-9a18-ad6765e0ca0a8761566173199177896.png"/><br><img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/d14651c9-4125-4d11-b654-3077ee3023d07933833805320852077.png"/> $\begin{align} \\ & \\ \text{Therefore, we get the number of pens as 10.}\end{align}$

 What is special about the graph of $4{{x}^{2}}+4x+1=0$ ?

 Plot the graph by substituting values of $x$ in the equation and finding values of $y$.

 $\begin{align} & 4{{x}^{2}}+4x+1=0 \\ & \text{ For }x = -1, y = 4 - 4 + 1 = 1 \\ & \text{ For }x = -0.5, y = 1 – 2 + 1 = 0 \\ & \text{ For }x = 0, y = 1 \\ & \text{ For }x = 0.5, y = 1 + 2 + 1 = 4 \\ & \text{ For }x = 1, y = 4 + 4 + 1 = 9\\ & \\ &\end{align}$<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/12d58ac6-0f2c-415a-82ab-ae6d03fe1ec02799732950756317240.png"/><img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/6bd2e1a8-8b93-4308-b198-9b839d8e70832923581521150312858.png"/> $\begin{align} \\ & \\ \text{There is only one x-intercept here.} \end{align}$

 Plot a graph for the equation $4{{x}^{2}}+6x+9=0$ and determine the nature of the roots.

 Find the values of $y$ for certain values of $x$ and plot the graph. Analyse the graph to determine the nature of roots.

 $\begin{align} & \text{We have }4{{x}^{2}}+6x+9=0.\\ & \text{We can take it as }y=4{{x}^{2}}+6x+9\\ & \text{Let us consider values of }x = -2, -1, 0, 1, 2\text{ and find values of “y”.} \\ & \text{For }x = -2,  y=4{{\left( -2 \right)}^{2}}+6\left( -2 \right)+9\\ & \Rightarrow y=13 \\ & \text{Similarly, filling the table, we have} \\ & \\ & \end{align}$<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a5e661ad-4794-45ac-ac7a-478c5d691cb27989304256858274033.png"/><img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/c67cae12-fba5-48e9-8ee9-a6e2d83341de2292088686327774972.png"/> $\begin{align} \\ & \\ \text{The graph doesn’t have x-intercepts. So, there are no real solutions.} \end{align}$