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This is the second of our many quizzes on Translating functions. Click on this to access more such competitive quizzes on Algebra.

Translating functions Quiz 2

mathematics

 What will be the root of the resulting function $g\left( x \right)$ when the given curve $f\left( x \right)$ is translated $5$ to the left? <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/cb5ccae3-02eb-491a-9f70-305d2f1855737470114789094868241.png"/>

 Find the function after translating the original function and then to find the root equate the function to $0$.

 $ \text{The root of the function }f\left( x \right)=x+2 \text{is -2}.\\ \text{When the function }f\left( x \right)=x+2 \text{ is translated 5 to the left} \\ \text{the equation of the function will be}\\ g\left( x \right)=x+7. \text{The root of this function will be,}\\ g\left( x \right)=x+7 \\ \text{when }g\left( x \right)=0, \\ x+7=0 \\ x=-7\text{ is the root of the function }g\left( x \right)\text{ }$

 What is the equation of the translated function of the graph of $f\left( x \right)={{x}^{3}}+4x+5$ from the given figure below? <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/2ec287ff-e423-43cd-87a9-9b48d8f94efc2027085293261748474.png"/>

  When a function is translated to the right then the new function can be found by replacing $x$ in the old function as $x$ minus the number of units that has been moved.

  $ \text{From the figure it is clear that the function } \\ f\left( x \right)={{x}^{3}}+4x+5 \text{has translated }\\ 7\text{ units to the right}.\\ \therefore \text{ we can replace }x\text{ in }f\left( x \right)\text{ as }x-7\text{ which will give us} \\ \text{ }g\left( x \right)={{\left( x-7 \right)}^{3}}+4\left( x-7 \right)+5 $

  What is the Y intercept of the graph of $y=\sqrt{x}+5$ after it is translated 4 units to the left.<img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/fb8575ea-1d74-4ddd-8f1b-06508bb1c1f48487069588056603298.png"/>

 Compare the translated function with the curve of the old function to find its behavior.

 $ \text{We know that, Y intercept is the point where} \\ \text{the curve of the function touches the Y-axis.}\\ \text{As it can be seen from the figure of the curve }y=\sqrt{x}+5,\\ \text{the Y-intercept will be 5. The curve can }\\ \text{be said to have starting point at }\left( 0,5 \right) \\ \text{and passing through }\left( 1,6 \right)\text{ and }\left( 4,7 \right).\\ \text{From this we can deduce that when the curve is} \\ \text{translated  4 units to the left , the equation of the }\\ \text{function will be }\sqrt{x+4}+5. \text{The starting point of this curve will be } \\ \left( -4,5 \right) \text{and it will pass} \\ \text{through }\left( -3,6 \right)\text{ and }\left( 0,7 \right).\\ \therefore \text{ 7 will be the Y-intercept of the function }\sqrt{x+4}+5$.

 Find the intersection between the graph of the function and its translated function. <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/6d228ee1-1303-4a47-8e35-e84f456e016a4177755498565900431.png"/>

 The point of intersection will be the point common to both the curves. So, to find it equate the two functions.

  &nbsp;$ \text{ Let the function be }f\left( x \right)={{x}^{2}}+2 \\ \text{and it is translated 4 units to the right and the }\\ \text{equation will be }g\left( x \right)={{\left( x-4 \right)}^{2}}+2. \\ \text{The intersection of }f\left( x \right)={{x}^{2}}+2 \text{ and }g\left( x \right)={{\left( x-4 \right)}^{2}}+2 \text{ is},\\ {{x}^{2}}+2={{\left( x-4 \right)}^{2}}+2\\ {{x}^{2}}+2={{x}^{2}}-8x+16+2 \\ {{x}^{2}}-8x+16+2-{{x}^{2}}-2=0 \\ -8x+16=0 \\ \therefore x=2 \\ \text{Putting }x=2\text{ in }f\left( x \right)\text{ we have,} \\ f\left( x \right)={{x}^{2}}+2 \\ f\left( 2 \right)={{2}^{2}}+2 \\ \text{        }=6 \\ \Rightarrow \text{ the point of intersection will be }\left( 2,6 \right).$

 What will be the product of distances between the point of intersection of the given graphs and their extreme points? <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/04bd82e7-11dd-42d5-82c2-510b5e97a6ac1786850757805125990.png"/>

 Point of intersection can be determined by equating the two functions and distance can be found by using the formula $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}-{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$.

$ \text{The point of intersection between the curves will be,}\\ 5{{x}^{2}}-5=5{{\left( x-4 \right)}^{2}}-5 \\ 5{{x}^{2}}-5=5{{x}^{2}}-40x+80-5 \\ 5{{x}^{2}}-40x+80-5-5{{x}^{2}}+5=0 \\ -40x+80=0 \\ \therefore x=2 \\ \text{Putting }x=2\text{ in }f\left( x \right)\text{ we have,} \\ f\left( x \right)=5{{x}^{2}}-5 \\ f\left( 2 \right)=5\left( {{2}^{2}} \right)-5 \\ \text{        }=15 \\ \Rightarrow \text{the point of intersection will be }\left( 2,15 \right). \\ \text{The extreme point of }f\left( x \right)=5{{x}^{2}}-5 \text{is }\left( 0,-5 \right)\text{and }g\left( x \right)=5{{\left( x-4 \right)}^{2}}-5 \text{is }\left( 4,-5 \right).\\ \text{The distance between point of intersection }\left( 2,15 \right) \text{and extreme point }\left( 0,-5 \right)\text{is,}\\ \sqrt{{{\left( 0-2 \right)}^{2}}-{{\left( -5-15 \right)}^{2}}} \\ \Rightarrow \sqrt{4+400}=\sqrt{404} \\ \Rightarrow 2\sqrt{101} \\ \text{The distance between point of intersection }\left( 2,15 \right) \text{and extreme point }\left( 4,-5 \right) \text{is,}\\ \sqrt{{{\left( 4-2 \right)}^{2}}-{{\left( -5-15 \right)}^{2}}} \\ \Rightarrow \sqrt{4+400}=\sqrt{404} \\ \Rightarrow 2\sqrt{101} \\ \text{Product of these distances will be }\left( 2\sqrt{101} \right)\left( 2\sqrt{101} \right)=404$.