Solve y=-(x-3)²-1: Vertical Shift of 5 Units

Which equation represents the the function

$y=-(x-3)^2-1$

moved 5 spaces up?

To solve this problem, we will follow these steps:

• Step 1: Identify the original function.
• Step 2: Apply the vertical shift of 5 units upward.
• Step 3: Write the resulting equation.

Let's go through each step:

Step 1: The given function is $y = -(x-3)^2 - 1$. This can be identified as a downward-facing parabola with its vertex at the point $(3, -1)$.

Step 2: To move the entire function 5 spaces up, we add 5 to the constant term $-1$ in the equation. The effect of this transformation is that the new vertex becomes $(3, -1 + 5) = (3, 4)$.

Step 3: Updating the function, we have:

$y = -(x-3)^2 - 1 + 5$

Simplify by combining the constants:

$y = -(x-3)^2 + 4$

This transformation results in the function moving 5 units up along the vertical axis to a new equation. The final equation is $y = -(x-3)^2 + 4$.

Therefore, the solution to the problem is $y=-(x-3)^2+4$, which is choice 4 from the given options.