Vertical Shift: Finding y=(x-2)² + 3 Through Function Transformation

Q: Why do we add +3 outside the parentheses and not inside?

Adding inside the parentheses like $ (x-2+3)^2 $ moves the graph horizontally, not vertically! To move up or down, always add or subtract outside the entire function.

Q: How can I remember the difference between up and down shifts?

Think logically: Moving up means higher y-values, so you add a positive number. Moving down means lower y-values, so you subtract or add a negative number.

Q: Does the shape of the parabola change when I shift it?

No! Vertical shifts are rigid transformations - they move the entire graph without changing its shape, width, or direction. Only the position changes.

Function Transformations with Vertical Shifts

Choose the equation that corresponds to the function

$y=(x-2)^2$

moved 3 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'll use the formula to shift the function

00:07 We want to shift 3 units horizontally upward, so we'll increase K

00:17 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 function

$y=(x-2)^2$

moved 3 spaces up.

Step-by-step solution

To solve this problem, we need to apply a vertical shift to the function $y = (x-2)^2$ .

When a function $y = f(x)$ is shifted vertically by a constant $k$ , the new function becomes $y = f(x) + k$ . In this problem, we need to shift the function three units up.

Given the original function $y = (x-2)^2$ :

Step 1: Identify the kind of transformation. We aim to move the function 3 spaces up.
Step 2: Add 3 to the existing equation. This yields $y = (x-2)^2 + 3$ .

The updated equation represents the translated parabola after shifting 3 units upwards.

Comparing this result with the given multiple-choice options, the correct corresponding equation is:

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Vertical Shift Rule: Add constant k to function: f(x) becomes f(x) + k
Technique: Moving up 3 units means adding +3: $y=(x-2)^2+3$
Check: Verify the vertex moved from (2,0) to (2,3) upward ✓

Common Mistakes

Avoid these frequent errors

Confusing vertical and horizontal shifts
Don't add the shift value inside the parentheses like (x-2+3)² = wrong position! This creates a horizontal shift instead of vertical. Always add the constant outside the function for vertical movement.

Practice Quiz

Test your knowledge with interactive questions

Which equation represents the function:

$$ y=x^2 $$

moved 2 spaces to the right

and 5 spaces upwards.

FAQ

Everything you need to know about this question

Why do we add +3 outside the parentheses and not inside?

Adding inside the parentheses like $(x-2+3)^2$ moves the graph horizontally, not vertically! To move up or down, always add or subtract outside the entire function.

How can I remember the difference between up and down shifts?

Think logically: Moving up means higher y-values, so you add a positive number. Moving down means lower y-values, so you subtract or add a negative number.

What happens to the vertex when I shift vertically?

The vertex moves straight up or down! Original vertex at (2,0) becomes (2,3) after shifting up 3 units. The x-coordinate stays the same, only y changes.

Does the shape of the parabola change when I shift it?

No! Vertical shifts are rigid transformations - they move the entire graph without changing its shape, width, or direction. Only the position changes.

How do I check if my transformed function is correct?

Pick any x-value and calculate y for both functions. The new y-value should be exactly 3 more than the original. For example: when x=2, original gives y=0, transformed gives y=3. Perfect!

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