Solving y=(x-1)² Vertical Shift: Finding the Downward Translation

Q: How do I remember which direction is up or down?

Easy rule: Adding a positive number moves the graph up, subtracting (or adding negative) moves it down. Think of it like a number line - positive goes up!

Q: Why does the answer have -3 instead of +3?

Because we're moving downward 3 units! Downward means subtract 3 from the y-values, giving us $ y = (x-1)^2 - 3 $.

Q: What's the difference between (x-3) and -3 in the equation?

$ (x-3) $ moves the parabola horizontally (left or right), while $ -3 $ at the end moves it vertically (up or down). They control different directions!

Q: How can I check if my translation is correct?

Pick a point on the original graph, like $ (1,0) $ from $ y=(x-1)^2 $. After moving down 3 units, it should become $ (1,-3) $. Test: $ y = (1-1)^2 - 3 = -3 $ ✓

Q: What if I need to move both up/down and left/right?

You can do both! For example: $ y = (x-2)^2 + 4 $ moves right 2 units and up 4 units. Just remember: inside parentheses = horizontal, outside = vertical.

Quadratic Functions with Vertical Translations

Which equation represents

$y=(x-1)^2$

moved units spaces downward?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the new function

00:04 We'll use the formula to shift the function

00:07 We want to shift 3 units horizontally downward, so we'll decrease 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

Which equation represents

$y=(x-1)^2$

moved units spaces downward?

Step-by-step solution

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

Step 1: Identify the given information
Step 2: Recognize the formula for vertical translation
Step 3: Apply the formula for a downward move
Step 4: Compare to multiple-choice answers to find a match

Now, let's work through each step:
Step 1: The original equation is $y = (x-1)^2$ .
Step 2: A translation "downward" results in the formula $y = (x-1)^2 + k$ , with $k = -3$ for moving 3 units downward.
Step 3: Substitute $k = -3$ into the formula to get $y = (x-1)^2 - 3$ .
Step 4: Among the choices provided, $y = (x-1)^2 - 3$ matches our formula for the translated equation.

Therefore, the solution to the problem is $y = (x-1)^2 - 3$ , corresponding to choice 2.

Final Answer

$y=(x-1)^2-3$

Key Points to Remember

Essential concepts to master this topic

Rule: Vertical shifts add or subtract constants outside the squared term
Technique: Downward movement subtracts: $y = (x-1)^2 - 3$
Check: Verify the constant is outside parentheses and has correct sign ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal and vertical shifts
Don't change the value inside the parentheses for vertical shifts = horizontal movement instead! Moving inside (x-1) affects left-right position, not up-down. Always add or subtract constants outside the parentheses for vertical translations.

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

How do I remember which direction is up or down?

Easy rule: Adding a positive number moves the graph up, subtracting (or adding negative) moves it down. Think of it like a number line - positive goes up!

Why does the answer have -3 instead of +3?

Because we're moving downward 3 units! Downward means subtract 3 from the y-values, giving us $y = (x-1)^2 - 3$ .

What's the difference between (x-3) and -3 in the equation?

$(x-3)$ moves the parabola horizontally (left or right), while $-3$ at the end moves it vertically (up or down). They control different directions!

How can I check if my translation is correct?

Pick a point on the original graph, like $(1,0)$ from $y=(x-1)^2$ . After moving down 3 units, it should become $(1,-3)$ . Test: $y = (1-1)^2 - 3 = -3$ ✓

What if I need to move both up/down and left/right?

You can do both! For example: $y = (x-2)^2 + 4$ moves right 2 units and up 4 units. Just remember: inside parentheses = horizontal, outside = vertical.

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