Transforming y=-x²: Shifting 5 Right and 4 Down

To solve this problem, we will apply transformations to the given quadratic function. Let's go through the solution step-by-step:

• Step 1: Understand the given function and desired transformations
• Step 2: Apply the horizontal shift
• Step 3: Apply the vertical shift
• Step 4: Write the transformed equation

Now, let's execute each step:
Step 1: The given function is $y = -x^2$. We need to transform this function by moving it 5 spaces to the right and 4 spaces downward. A horizontal shift involves modifying the $x$ term, whereas a vertical shift affects the $y$ values.
Step 2: To move the graph 5 spaces to the right, we replace $x$ with $x - 5$. This results in a new expression: $-(x-5)^2$. The $x-5$ indicates a shift to the right by 5 units.

Step 3: The function must also be moved downwards by 4 units. To achieve a vertical shift downward, we subtract 4 from the entire function. This means our function becomes $-(x-5)^2 - 4$. The $-4$ represents a downward shift of 4 units.

Step 4: Combining the horizontal and vertical shifts, the equation of the new function is: $y = -(x-5)^2 - 4$.

Therefore, the equation representing the function $y = -x^2$, after being moved 5 spaces to the right and 4 spaces downward, is $y = -(x-5)^2 - 4$.