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

Find the corresponding algebraic representation of the drawing:

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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