Quadratic Function (x-4)²: Shifting 2 Right and 3 Up

Q: Why does moving right make the number bigger inside the parentheses?

Think of it as compensation! When you move the parabola right, the x-values need to be larger to produce the same y-values. So $ (x-4)^2 $ becomes $ (x-6)^2 $ because now x must be 6 (not 4) to make the expression equal zero.

Q: What's the difference between horizontal and vertical shifts?

Horizontal shifts change what's inside the parentheses with x, while vertical shifts add or subtract outside the entire function. Moving right 2: (x-4) becomes (x-6). Moving up 3: add +3 to the whole expression.

Q: How do I remember which direction is which?

Use this trick: Horizontal shifts are backwards! To go right, subtract more. To go left, subtract less (or add). Vertical shifts are normal: up means +, down means -.

Q: Can I do the shifts in any order?

Yes! You can do horizontal and vertical shifts in any order and get the same final answer. They don't affect each other, so $ (x-6)^2 + 3 $ is the same whether you shift right first or up first.

Q: What if the original function had a different form?

The same rules apply! Whether it's $ (x-4)^2 $, $ x^2 $, or $ (x+1)^2 $, horizontal shifts always change what's with x inside parentheses, and vertical shifts add/subtract outside.

Quadratic Transformations with Combined Horizontal-Vertical Shifts

Choose which equation represents the function

$y=(x-4)^2$

moved 2 spaces to the right

and 3 spaces upwards upwards.

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose which equation represents the function

$y=(x-4)^2$

moved 2 spaces to the right

and 3 spaces upwards upwards.

Step-by-step solution

To solve this problem, we need to perform two transformations on the original function $y = (x-4)^2$ : a shift 2 units to the right and a shift 3 units upwards.

Step 1: Horizontal Shift (2 units to the right)
When a function is shifted to the right by $a$ , we replace $x$ with $x - a$ . In this case, $a = 2$ . Thus, replacing $x$ with $x-2$ in the original function $y = (x-4)^2$ results in $y = ((x-2)-4)^2 = (x-6)^2$ .

Step 2: Vertical Shift (3 units upwards)
To shift a function upwards by $k$ , add $k$ to the entire function. Here, $k = 3$ , so the transformed equation becomes $y = (x-6)^2 + 3$ .

Thus, the equation of the function after these transformations is $y = (x-6)^2 + 3$ .

The correct answer, as given in the problem, is indeed: $y = (x-6)^2 + 3$ . This corresponds to choice 4.

Final Answer

$y=(x-6)^2+3$

Key Points to Remember

Essential concepts to master this topic

Horizontal Rule: Right shift by h means replace x with (x-h)
Technique: From (x-4)² right 2: ((x-2)-4)² = (x-6)²
Check: Vertex moves from (4,0) to (6,3) confirming shifts ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal shift direction
Don't think moving right 2 changes (x-4) to (x-2) = moves left instead! Right shifts make the number inside parentheses MORE negative. Always remember: right shift by h means subtract h from the inside number: 4 + 2 = 6, giving (x-6)².

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 does moving right make the number bigger inside the parentheses?

Think of it as compensation! When you move the parabola right, the x-values need to be larger to produce the same y-values. So $(x-4)^2$ becomes $(x-6)^2$ because now x must be 6 (not 4) to make the expression equal zero.

What's the difference between horizontal and vertical shifts?

Horizontal shifts change what's inside the parentheses with x, while vertical shifts add or subtract outside the entire function. Moving right 2: (x-4) becomes (x-6). Moving up 3: add +3 to the whole expression.

How do I remember which direction is which?

Use this trick: Horizontal shifts are backwards! To go right, subtract more. To go left, subtract less (or add). Vertical shifts are normal: up means +, down means -.

Can I do the shifts in any order?

Yes! You can do horizontal and vertical shifts in any order and get the same final answer. They don't affect each other, so $(x-6)^2 + 3$ is the same whether you shift right first or up first.

What if the original function had a different form?

The same rules apply! Whether it's $(x-4)^2$ , $x^2$ , or $(x+1)^2$ , horizontal shifts always change what's with x inside parentheses, and vertical shifts add/subtract outside.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Parabola Families questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Quadratic Function (x-4)²: Shifting 2 Right and 3 Up

Quadratic Transformations with Combined Horizontal-Vertical Shifts

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does moving right make the number bigger inside the parentheses?

What's the difference between horizontal and vertical shifts?

How do I remember which direction is which?

Can I do the shifts in any order?

What if the original function had a different form?

🌟 Unlock Your Math Potential

More Questions

Parabola of the Form y=(x-p)²+k

Transforming y = x²: Shifting 4 Right and 3 Up

Calculate Negative Area: Finding Area Under y=-(x-4)² + 9

Transform y = x² : Shifting 2 Right and 5 Up - Finding the Equation

Transforming y=-x²: Shifting 5 Right and 4 Down