Questions: Consider the following function. t(x) = -(√(-x))/2 + 4 Identify the shape of the more basic function found in step 1.

Solution Steps

To identify the shape of the more basic function, we need to analyze the components of the given function \( t(x) = -\frac{\sqrt{-x}}{2} + 4 \). The basic function here is the square root function, \(\sqrt{x}\), which is a half-parabola opening to the right. However, due to the negative sign inside the square root, \(\sqrt{-x}\), the function reflects over the y-axis, creating a half-parabola opening to the left. The negative sign outside the square root, \(-\frac{\sqrt{-x}}{2}\), reflects it over the x-axis and scales it vertically by a factor of \(\frac{1}{2}\). Finally, the addition of 4 shifts the entire graph upwards by 4 units.

The given function is

\[ t(x) = -\frac{\sqrt{-x}}{2} + 4 \]

To identify the basic function, we need to consider the expression inside the square root, which is \(-x\). The basic function here is the square root function, \(\sqrt{x}\), but it is modified by a negative sign inside the square root, resulting in \(\sqrt{-x}\).

The basic function \(\sqrt{x}\) is a square root function, which typically has a shape that starts at the origin \((0,0)\) and increases gradually, forming a curve that rises to the right. However, in the function \(\sqrt{-x}\), the negative sign inside the square root reflects the graph across the y-axis. This means the graph of \(\sqrt{-x}\) starts at the origin and decreases to the left, forming a curve that falls to the left.

Final Answer