Question

The graph of \(y = \sqrt {{\bf{3}}x - {x^{\bf{2}}}} \) is given. Use transformations to create a function whose graph is as shown.

Short answer

The new function is \(y = 2\sqrt {7x - {x^2} - 10} \).

Step-by-step solution

Write observation for the transformed graph

From the transformed graph, the graph of the function \(f\left( x \right) = \sqrt {3x - {x^2}} \) is stretched vertically by the factor 2 and shifted 4 units to the right direction.

If \(y = f\left( x \right)\) is the function, then the new function becomes \(y = 2f\left( {x - 2} \right)\).

Obtain the equation for the transformed graph

The function describing thetransformed graphcan be expressed as shown below:

\(\begin{aligned}y &= 2f\left( {x - 2} \right)\\ &= 2\sqrt {3\left( {x - 2} \right) - {{\left( {x - 2} \right)}^2}} \\ &= 2\sqrt {3x - 6 - {x^2} + 4x - 4} \\ &= 2\sqrt {7x - {x^2} - 10} \end{aligned}\)

Thus, the equation of the transformed graph is \(y = 2\sqrt {7x - {x^2} - 10} \).