Chapter 1.3, Problem 7E

To determine

To create: A function from the given graph by using transformations.

Answer to Problem 7E

The function of the shown graph is $y=-\sqrt{3\left(x+4\right)-{\left(x+4\right)}^2}-1$.

Explanation of Solution

The equation of the given graph is $y=\sqrt{3x-x^2}$.

Let $y=f\left(x\right)$.

Notice that the given graph is a semi-circle with the domain [0, 3] and the range [0, 1.5].

Also, it is observed from the given graph that the graph is shifted 4 units to the left and shifted 1 unit downward.

Moreover, the graph is reflected about the origin.

Thus, add 4 to x as it shifted 4 units to the left; subtract 1fromf(x)as it shifted 1 unit downward and then multiply f(x) by −1 as the graph reflects about x-axis.

Therefore, the given function will change as $y=-f\left(x+4\right)-1$.

Thus, the function of the shown graph is $y=-\sqrt{3\left(x+4\right)-{\left(x+4\right)}^2}-1$.