Find y = x² Shifted Up 10 Units: Function Transformation Question

Q: Why do we add 10 outside the function instead of inside?

Adding outside the function affects the y-values directly. When you add 10 to $ x^2 $, every point on the graph moves up 10 units because you're adding 10 to the output!

Q: What would happen if I wrote y = (x+10)²?

That would move the parabola left 10 units, not up! Remember: changes inside the parentheses affect horizontal movement, changes outside affect vertical movement.

Q: How can I visualize this transformation?

Pick any point on $ y = x^2 $, like (2,4). After shifting up 10 units, this point becomes (2,14). The x-coordinate stays the same, but the y-coordinate increases by 10.

Q: What if the question asked to move it DOWN 10 units?

Then you would subtract 10: $ y = x^2 - 10 $. Moving down means subtracting from the function, moving up means adding to the function.

Q: Does this work for any function, not just parabolas?

Yes! Vertical shifts work the same way for any function. Whether it's linear, quadratic, cubic, or trigonometric - adding a constant moves the entire graph up by that amount.

Function Transformations with Vertical Shifts

Choose the equation that corresponds to the function

$y=x^2$

moved 10 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

00:12 We want to shift 10 units horizontally upward, so we'll increase K

00:25 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

$y=x^2$

moved 10 spaces up.

Step-by-step solution

To solve this problem, we must adjust the given function $y = x^2$ by moving it 10 units upwards. This transformation affects only the constant in the quadratic equation.

Let's consider the necessary steps:

Step 1: Recognize that the original function is $y = x^2$ .
Step 2: Calculate the new function after moving 10 units upwards by adding 10 to the original formula. Thus, our new equation becomes $y = x^2 + 10$ .
Step 3: Confirm this result aligns with one of the provided answer choices. Reviewing the choices, we see:
Choice 1: $y = x^2 + 10$
Choice 2: $y = x^2 - 10$
Choice 3: $y = x^2$
Choice 4: $y = 10x^2$
Step 4: Verify the correct choice is $y = x^2 + 10$ , which corresponds to moving the original function 10 units up.

Therefore, the equation of the function moved 10 spaces up is $y = x^2 + 10$ .

Final Answer

$y=x^2+10$

Key Points to Remember

Essential concepts to master this topic

Vertical Shift Rule: Adding to function moves graph up
Technique: For y = x² shifted up 10: y = x² + 10
Check: Test point (0,0) becomes (0,10) after transformation ✓

Common Mistakes

Avoid these frequent errors

Confusing vertical and horizontal shifts
Don't think y = (x+10)² moves the graph up 10 units = wrong direction! This actually moves the parabola LEFT 10 units. Always remember: add outside the function for vertical shifts UP.

Practice Quiz

Test your knowledge with interactive questions

Choose the equation that represents the function

$$ y=-x^2 $$

moved 3 spaces to the left

and 4 spaces up.

FAQ

Everything you need to know about this question

Why do we add 10 outside the function instead of inside?

Adding outside the function affects the y-values directly. When you add 10 to $x^2$ , every point on the graph moves up 10 units because you're adding 10 to the output!

What would happen if I wrote y = (x+10)²?

That would move the parabola left 10 units, not up! Remember: changes inside the parentheses affect horizontal movement, changes outside affect vertical movement.

How can I visualize this transformation?

Pick any point on $y = x^2$ , like (2,4). After shifting up 10 units, this point becomes (2,14). The x-coordinate stays the same, but the y-coordinate increases by 10.

What if the question asked to move it DOWN 10 units?

Then you would subtract 10: $y = x^2 - 10$ . Moving down means subtracting from the function, moving up means adding to the function.

Does this work for any function, not just parabolas?

Yes! Vertical shifts work the same way for any function. Whether it's linear, quadratic, cubic, or trigonometric - adding a constant moves the entire graph up by that amount.

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