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 \(y = - \sqrt { - {x^2} - 5x - 4} - 1\).

Step-by-step solution

Write the observation for the transformed graph

From the transformed graph, the graph of the function \(y = \sqrt {3x - {x^2}} \) is shifted 4 units to the left direction, so the function becomes \(y = \sqrt {3\left( {x + 4} \right) - {{\left( {x + 4} \right)}^2}} \).

Then this function reflected about the x-axis, so the function becomes \(y = - \sqrt {3\left( {x + 4} \right) - {{\left( {x + 4} \right)}^2}} \).

Then, it is shifted downward 1 unit, so the function becomes \(y = - \sqrt {3\left( {x + 4} \right) - {{\left( {x + 4} \right)}^2}} - 1\).

Obtain the equation for the transformed graph

The function describing the transformed graph can be expressed as shown below:

\(\begin{aligned}y &= - f\left( {x + 4} \right) - 1\\ &= - \sqrt {3\left( {x + 4} \right) - {{\left( {x + 4} \right)}^2}} - 1\\ &= - \sqrt {3x + 12 - \left( {{x^2} + 8x - 16} \right)} - 1\\ &= - \sqrt { - {x^2} - 5x - 4} - 1\end{aligned}\)

So, the equation of the transformed graph is \(y = - \sqrt { - {x^2} - 5x - 4} - 1\).