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This is one of our many quizzes on Graphing cube root functions. Click on this to access more such competitive quizzes on Number System.

Graphing cube root functions Quiz 2

mathematics

 In the below graph the black line is representing the equation of the function $y = \sqrt[3]{x}$ then find which equation the blue line is representing? $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/637249b7-68dd-4b19-be66-23ba1a5e2a047763889166013630859.png"/>

 Find parent function and then write the equation according to the shifting of the graph.

The graph is represented by the blue line is the replication of the graph represented by black line, therefore parent function of the required graph is $y = \sqrt[3]{x}$ $ \\ $ Also the graph represented by blue line is shifted $3$ units to the right and $4$ units vertically downward, therefore the required equation will be written as $ \\    \Rightarrow \left( {y + 4} \right) = \sqrt[3]{{\left( {x - 3} \right)}}   \\  \Rightarrow y = \sqrt[3]{{x - 3}} - 4   \\  $

 The domain and range of cube root function represented in the graph is $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/004d6364-b8d3-4599-859f-93274d1eb36b6073178131206647280.png"/>

 Try to find the roots from the graph and you have the roots then you have the factors of the equation too, so multiply all the three factors to get the required cube root equation and then find its domain and range.

 Since it is given that the given graph is a cube root function, and we have given all of the three roots of the cube root function, that is $x =  - 1,\;x = 2\;{\text{and}}\;x = 3$  $ \\ $ So to find the equation, $ y = \left( {x + 1} \right)\left( {x - 2} \right)\left( {x - 3} \right)   \\   \Rightarrow y = {x^3} - 4{x^2} + x + 6   \\  $ Since we get a polynomial function, therefore the domain and range of the given function will be all real values.

 Equation of the given function will be $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/e94123d4-07bb-40ee-8784-ddb2bf8451181526007930589023876.png"/>

 Observe the points on the graph and use the derivative tests to check for the right equation.

The graph cuts the x-axis at $x = 15$. $ \\ $ So one condition is at $x = 15$, $y = 0$. $ \\ $ For the y-intercept, at $x = 0$, $y =  - 9$. $ \\ $ These conditions are satisfied by the function given in option $(b)$. $ \\ $ We can also check the other conditions such as the derivative test to verify that option $(b)$ is the correct answer. $ \\ $ The curve is flattened only once so the first derivation of the function will be zero at only one value of $x$. $ \\  \dfrac{{d\left( {{{\left( {\dfrac{x}{5} - 2} \right)}^3} - 1} \right)}}{{dx}} = 0   \\   \Rightarrow 3{\left( {\dfrac{x}{5} - 2} \right)^2}.\dfrac{1}{5} = 0   \\   \Rightarrow x = 10   \\  $ We observe that it is similar to what is given in the graph. $ \\ $ Checking the second derivative, $  \dfrac{{d\left( {\dfrac{3}{5}{{\left( {\dfrac{x}{5} - 2} \right)}^2}} \right)}}{{dx}}   \\   \Rightarrow \dfrac{6}{5}\left( {\dfrac{x}{5} - 2} \right)   \\  $ The second derivative is positive for $x > 10$ and correspondingly the curve is concave up in this region. $ \\ $ The second derivative is negative for $x < 10$ and correspondingly the curve is concave down in this region.

 Graph of the equation $y = 7{x^3} + 3$ can be drawn by

 Multiplication of $x$ with a constant greater than one caused shrinking of the graph.

Given $y = 7{x^3} + 3 \Leftrightarrow \left( {y - 3} \right) = 7{x^3}$ and its parent equation is $y = {x^3}$ $ \\ $ Multiply constant with $x$ then graph shrinks, given that constant should be greater than one and subtract a constant from $y$ the graph shifts vertically upward $ \\ $ Therefore the graph of $y = 7{x^3} + 3$ can be drawn by shrinking the graph of $y = {x^3}$ seven times and shifting it $3$ units up.

 Graph of the equation $2y =  - 3{x^3} + 4$ is

 Simplify the equation by making $y$ dependent variable and then try to draw the graph by transforming the graph of the parent function of the given function.

We have $ 2y =  - 3{x^3} + 4   \\   \Rightarrow y = \dfrac{{ - 3{x^3} + 4}}{2}   \\   \Rightarrow y = \dfrac{{ - 3{x^3}}}{2} + 2   \\  $ The parent function of the given function is $y = {x^3}$ $ \\ $ So its graph is $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/a1269f6a-6cfc-4f35-9cdf-6f071995ea0a5447839453589240871.png"/> $ \\ $ Writing the equation as $y =  - {x^3}$ $ \\ $ Negative sign will invert the graph with respect to the y-axis as follows $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/97aaab52-7bf0-4b52-b3e1-4da1d45bf2f6622712911707443421.png"/> $ \\ $ Multiplying it with $\dfrac{3}{2}$ $ \\ y =  - \dfrac{3}{2}{x^3} \\ $ It will shrink the graph by $\dfrac{3}{2}$ times as follows $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/26029c4c-aa98-442c-b5f3-db101527d4034604971695824756417.png"/> $ \\ $ Adding $2$ which will shift the graph $2$ units vertically upward as follows $ \\ $ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/cecf0639-788d-4cf7-b964-a936c5caf5533261395186084947116.png"/> $ \\ $ Hence, the required graph for the given equation.