Vertical Shift: Find y = -(x-6)² + 4 After Translation

Q: How do I know if it's moving up or down?

Moving up: Add a positive number outside the function. Moving down: Add a negative number (or subtract). The phrase "4 spaces up" means add +4 to the entire function.

Q: Why doesn't the negative sign in front change?

The negative sign determines the parabola's direction (opens up or down). Vertical shifts only move the graph up/down without changing its shape or orientation, so the negative stays!

Q: What happens to the vertex when I shift vertically?

Only the y-coordinate of the vertex changes! Original vertex: (6,0). After moving up 4 units: (6,4). The x-coordinate stays the same.

Q: Can I have multiple transformations at once?

Yes! You can combine horizontal shifts (inside parentheses), vertical shifts (outside), and reflections (negative signs) all in one equation like $ y = -(x-3)^2 + 5 $.

Q: How is this different from stretching or compressing?

Stretching/compressing multiplies the function by a number, changing its width. Shifting adds/subtracts a number, moving the entire graph without changing its shape.

Parabola Transformations with Vertical Translations

Choose the equation that corresponds to the the function

$y=-(x-6)^2$

moved 4 spaces up.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the new function

00:03 We will use the formula to shift the function

00:09 We want to shift 4 units horizontally upward, so we'll increase K

00:18 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the equation that corresponds to the the function

$y=-(x-6)^2$

moved 4 spaces up.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the original function.
Step 2: Apply the vertical transformation.
Step 3: Confirm the new function equation matches a given choice.

Now, let's work through each step:
Step 1: The initial function is $y = -(x-6)^2$ . This represents a parabola that opens downward, vertex at (6,0).
Step 2: To shift the graph of the function 4 units up, we add 4 to the entire function:
$y = -(x-6)^2 + 4$ .
Step 3: Review the provided choices to find the match:
- Choice 4: $y=-(x-6)^2+4$ .
This matches our transformation result.

Therefore, the solution to the problem is $y = -(x-6)^2 + 4$ .

Final Answer

$y=-(x-6)^2+4$

Key Points to Remember

Essential concepts to master this topic

Transformation Rule: Add constant outside function for vertical shift up
Technique: $y = -(x-6)^2$ becomes $y = -(x-6)^2 + 4$
Check: New vertex moves from (6,0) to (6,4) when shifted up 4 ✓

Common Mistakes

Avoid these frequent errors

Adding constant inside the parentheses instead of outside
Don't write y = -(x-6+4)² = y = -(x-2)² which creates horizontal shift! This moves the parabola left/right instead of up/down. Always add the vertical shift constant outside the squared term: y = -(x-6)² + 4.

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

How do I know if it's moving up or down?

Moving up: Add a positive number outside the function. Moving down: Add a negative number (or subtract). The phrase "4 spaces up" means add +4 to the entire function.

Why doesn't the negative sign in front change?

The negative sign determines the parabola's direction (opens up or down). Vertical shifts only move the graph up/down without changing its shape or orientation, so the negative stays!

What happens to the vertex when I shift vertically?

Only the y-coordinate of the vertex changes! Original vertex: (6,0). After moving up 4 units: (6,4). The x-coordinate stays the same.

Can I have multiple transformations at once?

Yes! You can combine horizontal shifts (inside parentheses), vertical shifts (outside), and reflections (negative signs) all in one equation like $y = -(x-3)^2 + 5$ .

How is this different from stretching or compressing?

Stretching/compressing multiplies the function by a number, changing its width. Shifting adds/subtracts a number, moving the entire graph without changing its shape.

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