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

Q: How do I remember which direction is which?

Use this memory trick: (x-h) moves right h units, (x+h) moves left h units. The sign is always opposite to the direction you want to move!

Q: What happens to the negative sign in front of x²?

The negative sign stays with the function! It affects the shape (opens downward), not the position. Transformations only change the x and y parts: $ y = -(x-5)^2 - 4 $

Q: Can I check my answer by graphing?

Absolutely! The vertex of the original function y = -x² is at (0,0). After transformations, it should be at (5,-4). If your equation gives this vertex, you're correct!

Q: What if I need to move left or up instead?

For moving left h units: use (x+h). For moving up k units: add k. Example: left 3, up 2 would give $ y = -(x+3)^2 + 2 $

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Which equation represents the function:

$y=-x^2$

when moved 5 spaces to the right

and 4 spaces horizontally and downward?

2

Step-by-step solution

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 $y = -(x-5)^2 - 4$ .

3

Final Answer

$y=-(x-5)^2-4$

Key Points to Remember

Essential concepts to master this topic

Rule: Right h units means replace x with (x-h), down k units means subtract k
Technique: For y = -x² moved right 5, down 4: y = -(x-5)² - 4
Check: Test vertex shift: original (0,0) becomes (5,-4) in transformed function ✓

Common Mistakes

Avoid these frequent errors

Confusing direction signs for horizontal shifts
Don't write (x+5) when moving right 5 units = moves left instead! The sign is opposite to the direction. Always use (x-h) for moving right h units and (x+h) for moving left h units.

Practice Quiz

Test your knowledge with interactive questions

Find the corresponding algebraic representation of the drawing:

FAQ

Everything you need to know about this question

Why is it (x-5) when moving right, not (x+5)?

+

Think of it this way: to get the same y-value as before, x must be 5 larger. So when x=5 in the new function, (x-5) = 0, giving the same result as x=0 in the original function!

How do I remember which direction is which?

+

Use this memory trick: (x-h) moves right h units, (x+h) moves left h units. The sign is always opposite to the direction you want to move!

What happens to the negative sign in front of x²?

+

The negative sign stays with the function! It affects the shape (opens downward), not the position. Transformations only change the x and y parts: $y = -(x-5)^2 - 4$

Can I check my answer by graphing?

+

Absolutely! The vertex of the original function y = -x² is at (0,0). After transformations, it should be at (5,-4). If your equation gives this vertex, you're correct!

What if I need to move left or up instead?

+

For moving left h units: use (x+h). For moving up k units: add k. Example: left 3, up 2 would give $y = -(x+3)^2 + 2$

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

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