Translate y=-(x-2)²+4: Moving a Quadratic Function 10 Units Down

Quadratic Transformations with Vertical Translation

Which equation represents the function:

y=−(x−2)2+4 y=-(x-2)^2+4

moved 10 spaces down?

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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
00:10 We want to shift 10 units horizontally downward, so we'll decrease K
00:26 And this is the solution to the question

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

moved 10 spaces down?

2

Step-by-step solution

To solve this problem, we will perform the following steps:

  • Step 1: Identify the original function. It is y=−(x−2)2+4 y = -(x-2)^2 + 4 .
  • Step 2: Determine how many units to move the function. According to the problem, we move it 10 spaces down, which means we subtract 10 from the entire function.
  • Step 3: Perform the vertical transformation by modifying the constant term. The new function is:

y=−(x−2)2+4−10 y = -(x-2)^2 + 4 - 10 .

Step 4: Simplify the resulting expression:

y=−(x−2)2−6 y = -(x-2)^2 - 6 .

This adjusted equation shows the original parabola moved 10 spaces down.

If we look at the given choices, our result corresponds to choice 3.

Therefore, the equation representing the function moved 10 spaces down is y=−(x−2)2−6 y = -(x-2)^2 - 6 .

3

Final Answer

y=−(x−2)2−6 y=-(x-2)^2-6

Key Points to Remember

Essential concepts to master this topic
  • Vertical Translation: Add or subtract constant to move parabola up/down
  • Technique: Moving down 10 means subtract 10: y=−(x−2)2+4−10 y = -(x-2)^2 + 4 - 10
  • Check: New y-intercept should be 10 units lower than original ✓

Common Mistakes

Avoid these frequent errors
  • Changing the wrong part of the equation
    Don't modify the coefficient or the (x-2)² term when translating vertically = wrong shape and position! Vertical translation only affects the constant term. Always add/subtract from the constant at the end of the equation.

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 don't I change the -1 coefficient or the (x-2)² part?

+

The coefficient -1 controls if the parabola opens up or down, and (x-2)² controls horizontal position. Only the constant term controls vertical position!

How do I remember if moving down means add or subtract?

+

Think logically: moving down means smaller y-values, so you subtract. Moving up means larger y-values, so you add.

What's the difference between moving the whole function vs just changing one part?

+

Moving the whole function means every point moves the same direction. Changing just one part (like the coefficient) would change the shape, not just the position.

How can I check if my transformed equation is correct?

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Pick any x-value and calculate y for both equations. The new y-value should be exactly 10 less than the original y-value!

Does the vertex move too when I translate vertically?

+

Yes! The vertex of y=−(x−2)2+4 y = -(x-2)^2 + 4 is at (2, 4). After moving down 10, it's at (2, -6).

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