Analyzing (x+4)² vs x²: Understanding Horizontal Shifts in Quadratic Functions

Q: Why does adding 4 make the graph go left instead of right?

Think about it this way: in $ (x+4)^2 $, when does the expression equal zero? When x = -4, not x = 4! This means the vertex (lowest point) moved from x = 0 to x = -4, which is 4 units to the left.

Q: How is this different from y = x² + 4?

$ y = x^2 + 4 $ adds 4 outside the parentheses, shifting the graph up 4 units. $ y = (x+4)^2 $ adds 4 inside the parentheses, shifting left 4 units. Inside = horizontal, outside = vertical!

Q: What would (x-4)² look like?

$ y = (x-4)^2 $ would shift the graph right 4 units! The vertex would be at (4,0) because x = 4 makes the expression equal zero.

Q: How can I remember the direction of horizontal shifts?

Use this trick: "Opposite Day for Horizontal!" Whatever sign you see inside the parentheses, the shift goes the opposite direction. Plus sign = left shift, minus sign = right shift.

Q: Does the shape of the parabola change?

No! $ y = (x+4)^2 $ has exactly the same shape as $ y = x^2 $. Only the position changes - it's like sliding the entire graph horizontally without stretching or flipping it.

Horizontal Transformations with Function Notation

$y=(x+4)^2$ is the function $y=x^2$ moved left 4 spaces.

❤️ 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

00:00 Is the function the same function 4 steps to the left?

00:03 Let's observe at the origin point where K and P equal 0

00:11 Let's substitute in the quadratic function formula and solve

00:19 Now we want to take 4 steps left, meaning subtract 4 from P

00:24 Let's substitute in the quadratic function formula and solve

00:44 Negative times negative always equals positive

00:47 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

$y=(x+4)^2$ is the function $y=x^2$ moved left 4 spaces.

Step-by-step solution

To solve this problem, we'll determine how the transformation $y = (x+4)^2$ affects the graph of $y = x^2$ .

The function $y = x^2$ is a standard parabola centered at the origin.

In $y = (x+4)^2$ , the positive number inside the parentheses indicates a transformation that moves the entire graph horizontally.

Specifically, the expression $(x + 4)$ means we take the original $x$ value and add 4 to it before squaring, effectively shifting the graph.

According to properties of horizontal translations, when you add a positive number to $x$ inside the function—here, the +4 in $(x+4)$ —the graph of the function $y = x^2$ shifts 4 units to the left along the x-axis.

Therefore, the transformation described by $y = (x+4)^2$ is indeed the graph of $y = x^2$ moved 4 spaces to the left.

Thus, the correct answer to this problem is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: Adding inside parentheses shifts the graph left
Technique: $(x+4)^2$ means shift $x^2$ left 4 units
Check: Vertex moved from (0,0) to (-4,0) confirms leftward shift ✓

Common Mistakes

Avoid these frequent errors

Thinking +4 means shift right
Don't assume (x+4) shifts right because of the plus sign = wrong direction! The +4 inside parentheses actually creates a leftward shift because x must be 4 units smaller to make the expression zero. Always remember: inside parentheses, + means left and - means right.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

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

With the X

FAQ

Everything you need to know about this question

Why does adding 4 make the graph go left instead of right?

Think about it this way: in $(x+4)^2$ , when does the expression equal zero? When x = -4, not x = 4! This means the vertex (lowest point) moved from x = 0 to x = -4, which is 4 units to the left.

How is this different from y = x² + 4?

$y = x^2 + 4$ adds 4 outside the parentheses, shifting the graph up 4 units. $y = (x+4)^2$ adds 4 inside the parentheses, shifting left 4 units. Inside = horizontal, outside = vertical!

What would (x-4)² look like?

$y = (x-4)^2$ would shift the graph right 4 units! The vertex would be at (4,0) because x = 4 makes the expression equal zero.

How can I remember the direction of horizontal shifts?

Use this trick: "Opposite Day for Horizontal!" Whatever sign you see inside the parentheses, the shift goes the opposite direction. Plus sign = left shift, minus sign = right shift.

Does the shape of the parabola change?

No! $y = (x+4)^2$ has exactly the same shape as $y = x^2$ . Only the position changes - it's like sliding the entire graph horizontally without stretching or flipping it.

🌟 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